{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":49718,"title":"Malus’ Law (theta) between polarizations","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 20.736px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.5px 10.368px; transform-origin: 406.5px 10.368px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.496px 10.368px; text-align: left; transform-origin: 383.502px 10.368px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the angle (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eθ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e theta) ,I1 and I2 will be given \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = t(x,y)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 10;y=5\r\ny_correct = 60;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=5\r\ny_correct = 75.5225;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=10\r\ny_correct = 60;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=4\r\ny_correct = 78.4630;\r\nassert(isequal(t(x,y),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":610936,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-31T03:26:31.000Z","updated_at":"2026-04-02T12:26:44.000Z","published_at":"2020-12-31T03:44:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the angle (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e theta) ,I1 and I2 will be given \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61185,"title":"Compute wheel slip ratio during braking.","description":"During braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as wheel slip ratio, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\r\nGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 383px 31.5px; text-align: left; transform-origin: 383px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eDuring braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ewheel slip ratio\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 383px 21px; text-align: left; transform-origin: 383px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = wheelSlipRatio(v_vehicle, v_wheel)\r\ns = 0;\r\nend\r\n","test_suite":"%%\r\nv_vehicle = 20; v_wheel = 18;\r\ns_correct = 0.1;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n%%\r\nv_vehicle = 25; v_wheel = 0;\r\ns_correct = 1;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n%%\r\nv_vehicle = 0; v_wheel = 0;\r\ns_correct = 0;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":2305225,"edited_by":2305225,"edited_at":"2026-02-02T06:36:51.000Z","deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-02-02T06:36:05.000Z","updated_at":"2026-04-04T03:41:48.000Z","published_at":"2026-02-02T06:36:51.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDuring braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ewheel slip ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59711,"title":" Calculating distance of lightning based on time delay","description":"If we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 21px; transform-origin: 407px 21px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIf we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = your_fcn_name(t)\r\n  %Easier than it looks\r\nend","test_suite":"%%\r\nt = 1;\r\ny_correct = 343;\r\nassert(isequal(your_fcn_name(t),y_correct))\r\n%%\r\nt = 57;\r\ny_correct = 19551;\r\nassert(isequal(your_fcn_name(t),y_correct))\r\n%%\r\nt = randi(1000);\r\ny_correct = 343*t;\r\nassert(isequal(your_fcn_name(t),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":4207616,"edited_by":4207616,"edited_at":"2024-03-23T10:18:40.000Z","deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2024-03-23T10:18:40.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-23T10:15:53.000Z","updated_at":"2025-08-26T12:10:35.000Z","published_at":"2024-03-23T10:18:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42689,"title":"Distance a ball travels after throwing vertically","description":"Calculate the total distance *'d'* (in meters) a ball would travel after *'s'* seconds and starting velocity of *'v'* (in m/s). \"Please note that its not relative distance but total distance travelled\"\r\n\r\ngravity=10 m/s^2\r\n\r\nExample: initial speed = +20, how long does the ball travel after 2.5 sec\r\nd=21.25","description_html":"\u003cp\u003eCalculate the total distance \u003cb\u003e'd'\u003c/b\u003e (in meters) a ball would travel after \u003cb\u003e's'\u003c/b\u003e seconds and starting velocity of \u003cb\u003e'v'\u003c/b\u003e (in m/s). \"Please note that its not relative distance but total distance travelled\"\u003c/p\u003e\u003cp\u003egravity=10 m/s^2\u003c/p\u003e\u003cp\u003eExample: initial speed = +20, how long does the ball travel after 2.5 sec\r\nd=21.25\u003c/p\u003e","function_template":"function d = distance(s,v)\r\nif v-10*s\u003e0\r\nd=((v-s*10)+v)/2*s\r\nelseif 2*v-10*s\u003c0\r\nd=v*v/10\r\nelse\r\nd=v/10*v/2+(s-v/10)^2*10/2\r\nend\r\nend","test_suite":"%%\r\nv = 20;\r\ns=2;\r\nd = 20;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv = 20;\r\ns=2.5;\r\nd = 21.25;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv = 20;\r\ns=12.5;\r\nd=40;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv=50;\r\ns=10;\r\nd=250;\r\n\r\nv=5;\r\ns=5;\r\nd=2.5;\r\nassert(isequal(distance(s,v),d));","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":9199,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":57,"test_suite_updated_at":"2015-11-25T15:32:59.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2015-11-25T11:24:15.000Z","updated_at":"2026-02-04T17:51:38.000Z","published_at":"2015-11-25T15:29:05.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the total distance\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e'd'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (in meters) a ball would travel after\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e's'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e seconds and starting velocity of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e'v'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (in m/s). \\\"Please note that its not relative distance but total distance travelled\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003egravity=10 m/s^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: initial speed = +20, how long does the ball travel after 2.5 sec d=21.25\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":48685,"title":"Laws of motion 3","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the final velocity of an object with initial velocity 'u', acceleration 'a' and distance travelled 's'. Round to nearest number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function v = your_fcn_name(u,a,s)\r\n  y = x;\r\nend","test_suite":"%%\r\nu=0;\r\na=1;\r\ns=1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=0;\r\na=1;\r\ns=10;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=20;\r\na=2;\r\ns=10;\r\ny_correct = 21;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=0;\r\na=1;\r\ns=7;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n","published":true,"deleted":false,"likes_count":24,"comments_count":5,"created_by":644918,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3285,"test_suite_updated_at":"2020-12-21T16:57:51.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-21T16:53:43.000Z","updated_at":"2026-04-04T03:59:26.000Z","published_at":"2020-12-21T16:53:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the final velocity of an object with initial velocity 'u', acceleration 'a' and distance travelled 's'. Round to nearest number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":441,"title":"Drying sweater?","description":"* A sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes. \r\n* Veronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer. \r\n* How many minutes she should wait so that the sweater weighs w times its dry weight? \r\n* Please try a general solution, the test suite may expand later.","description_html":"\u003cul\u003e\u003cli\u003eA sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes.\u003c/li\u003e\u003cli\u003eVeronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer.\u003c/li\u003e\u003cli\u003eHow many minutes she should wait so that the sweater weighs w times its dry weight?\u003c/li\u003e\u003cli\u003ePlease try a general solution, the test suite may expand later.\u003c/li\u003e\u003c/ul\u003e","function_template":"function m = drying(x,w)\r\n   m=x/w;\r\nend","test_suite":"%%\r\nx=2; w=1.5; m=drying(x,w);\r\nm_correct = 69;\r\nassert(round(m)==m_correct)\r\n%%\r\nx=3; w=2; m=drying(x,w);\r\nm_correct = 69;\r\nassert(round(m)==m_correct)\r\n%%\r\nx=5; w=2; m=drying(x,w);\r\nm_correct = 139;\r\nassert(round(m)==m_correct)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":166,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":"2012-03-06T17:52:02.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-02T17:09:51.000Z","updated_at":"2025-05-13T14:59:54.000Z","published_at":"2012-03-13T16:01:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eVeronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHow many minutes she should wait so that the sweater weighs w times its dry weight?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease try a general solution, the test suite may expand later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":480,"title":"Aufbau principle ","description":"Given the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\r\nsum(x)==sumofx\r\nlength of x is the shortest possible\r\nx has positive integers only\r\nif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\r\nif x(k+1)\u003e0 then x(k) must be maximum possible.\r\nFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info: Aufbau Principle, Electron Shell.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 216.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 108.083px; transform-origin: 407px 108.083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363.5px 8px; transform-origin: 363.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 51.0833px; transform-origin: 391px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 51px 8px; transform-origin: 51px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esum(x)==sumofx\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107px 8px; transform-origin: 107px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elength of x is the shortest possible\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ex has positive integers only\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 246px 8px; transform-origin: 246px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 150.5px 8px; transform-origin: 150.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eif x(k+1)\u0026gt;0 then x(k) must be maximum possible.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eAufbau Principle\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eElectron Shell\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function electrons = Aufbau(sumofx)\r\nif sumofx==3; \r\n   x=[2 1]; \r\n   electrons='1s2,2s1'; \r\nend\r\n","test_suite":"%%\r\nsumofx = 3; % Lithium\r\nelectrons = '1s2,2s1';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 6; % Carbon\r\nelectrons = '1s2,2s2,2p2';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 10; % Neon\r\nelectrons = '1s2,2s2,2p6';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 17; % Chlorine\r\nelectrons = '1s2,2s2,2p6,3s2,3p5';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 20; % Chlorine\r\nelectrons = '1s2,2s2,2p6,3s2,3p6,4s2';\r\nassert(isequal(electrons,Aufbau(sumofx)))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":166,"edited_by":223089,"edited_at":"2022-10-29T14:54:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2022-10-29T14:54:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-12T03:36:58.000Z","updated_at":"2026-01-02T13:13:55.000Z","published_at":"2012-03-12T15:44:42.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esum(x)==sumofx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003elength of x is the shortest possible\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex has positive integers only\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif x(k+1)\u0026gt;0 then x(k) must be maximum possible.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eAufbau Principle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eElectron Shell\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44936,"title":"Float like a cannonball","description":"Given gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative).\r\nHint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!","description_html":"\u003cp\u003eGiven gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative).\r\nHint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!\u003c/p\u003e","function_template":"function s = CannonBall(u,theta)\r\n%Stones taught me to fly...\r\n  s = u*theta;\r\nend","test_suite":"%%\r\nu= 100;\r\ntheta=85;\r\ny_correct = 177;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 31.42;\r\ntheta=45;\r\ny_correct = 101;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 31.42;\r\ntheta=-41;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=30;\r\ny_correct = 883;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= -100;\r\ntheta=30;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n\r\n%%\r\nu= -100;\r\ntheta=210;\r\ny_correct = 883;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n\r\n%%\r\nu= 100;\r\ntheta=210;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 0;\r\ntheta=40;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=90;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=0;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nfiletext = fileread('CannonBall.m');\r\nassert(isempty(strfind(filetext, 'regexp'))); assert(isempty(strfind(filetext, 'eval'))) \r\n\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":170350,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2019-08-02T09:56:25.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2019-08-02T09:45:02.000Z","updated_at":"2026-01-02T13:04:50.000Z","published_at":"2019-08-02T09:45:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative). Hint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":47435,"title":"Wavy gravity","description":"In a parallel universe the gravity works very strangely.\r\nIndeed, gravity is equal to: g = sin( (pi/2)*t/60 ) [m/s^2], where t is the time expressed in seconds, and it acts along the direction normal to the ground.\r\nAt time t = 0 seconds, a ball is at height y = 1 meter and has zero velocity. What is the position y (meters) of the ball as a function of time (seconds)?","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.5px 8px; transform-origin: 168.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn a parallel universe the gravity works very strangely.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIndeed, gravity is equal to: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eg\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105.5px 8px; transform-origin: 105.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = sin( (pi/2)*t/60 ) [m/s^2], where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003et\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.5px 8px; transform-origin: 174.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the time expressed in seconds, and it acts along the direction normal to the ground.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.5px 8px; transform-origin: 23.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003et\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0 seconds, a ball is at height \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168px 8px; transform-origin: 168px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 1 meter and has zero velocity. What is the position \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79px 8px; transform-origin: 79px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (meters) of the ball as a function of time (seconds)?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function yf = parUnig(tf)\r\ny = [];\r\nend","test_suite":"%%\r\ntf = 30; % seconds\r\ny_correct = 115; % meters\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n\r\n%%\r\ntf = 60;\r\ny_correct = 834;\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n\r\n%%\r\ntf = 120;\r\ny_correct = 4585;\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":280347,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":"2022-01-01T10:27:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-11-10T11:04:29.000Z","updated_at":"2022-01-01T10:27:26.000Z","published_at":"2020-11-10T11:06:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn a parallel universe the gravity works very strangely.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIndeed, gravity is equal to: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = sin( (pi/2)*t/60 ) [m/s^2], where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the time expressed in seconds, and it acts along the direction normal to the ground.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 0 seconds, a ball is at height \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 1 meter and has zero velocity. What is the position \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (meters) of the ball as a function of time (seconds)?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46968,"title":"Electric Potential Energy","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 132px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 66px; transform-origin: 407px 66px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCalculate the magnitude of Electric potential energy between two given charges [q1, q2] with spatial coordinates in the form [x1 y1; x2 y2]. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTake the value of absolute dielectric permittivity 1e-9/(36pi).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAll units are SI units.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eUse of if-else is prohibited to prevent hard-coded solutions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function F = ESF(q, c)\r\nq,c;\r\nend","test_suite":"%%\r\nfiletext = fileread('ESF.m');\r\nassert(isempty(strfind(filetext, 'if ')))\r\n\r\n%%\r\nq = [1e-5 1e-5];\r\nc = [1 1; 4 5];\r\n\r\nassert(abs(ESF(q,c)-0.18)\u003c1e-6);\r\n\r\n%%\r\nq = [1e-7 1e-6];\r\nc = [0 0; -6 -8];\r\n\r\nassert(abs(ESF(q,c)-9e-5)\u003c1e-6);\r\n\r\n%%\r\nq = [1e-2 -1e-3];\r\nc = [0.5 0.75; -2.5 -2.25]/sqrt(2);\r\n\r\nassert(abs(ESF(q,c)+3e4)\u003c1e-6);\r\n\r\n%%\r\nq = [-1 -1];\r\nc = [-3 -4;-3 -1];\r\n\r\nassert(abs(ESF(q,c)-3e9)\u003c1e-6);\r\n\r\n%%\r\nq = [1e5 1e-15];\r\nc = [-0 +0; -200 990];\r\n\r\nassert(abs(ESF(q,c)-8.911e-4)\u003c1e-6);\r\n\r\n%%\r\nq = [-1e-7 1e-13];\r\nc = [-27 100; 92 -220];\r\n\r\nassert(abs(ESF(q,c)+2.6361e-13)\u003c1e-6);\r\n\r\n%%\r\nq = [1e30 1e-30];\r\nc = [-exp(1) pi; -exp(1) pi+1];\r\n\r\nassert(abs(ESF(q,c)-9e9)\u003c1e-6);","published":true,"deleted":false,"likes_count":2,"comments_count":9,"created_by":223089,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":"2020-12-27T17:53:49.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2020-10-20T11:19:48.000Z","updated_at":"2025-04-02T17:27:01.000Z","published_at":"2020-10-20T11:19:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the magnitude of Electric potential energy between two given charges [q1, q2] with spatial coordinates in the form [x1 y1; x2 y2]. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTake the value of absolute dielectric permittivity 1e-9/(36pi).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll units are SI units.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUse of if-else is prohibited to prevent hard-coded solutions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54710,"title":"Compute the period of a pendulum started from a finite initial angle","description":"Cody Problem 49830 asks for the period  of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle  that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \r\nWrite a function that takes the initial angle and returns , where  is the length of the pendulum and  is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49830\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 49830\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for the period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"theta0\" style=\"width: 15px; height: 20px;\" width=\"15\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the initial angle and returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T\\sqrt{g/L}\" style=\"width: 52.5px; height: 20.5px;\" width=\"52.5\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the length of the pendulum and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eg\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2pi\" style=\"width: 19.5px; height: 18px;\" width=\"19.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pendulumPeriod(theta0)\r\n  T = theta0-theta0^3/3!+theta0^5/5!+higher order terms;\r\nend","test_suite":"%%\r\nth = pi/7;\r\nT_correct = 6.363207946270837;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/5;\r\nT_correct = 6.44181661515865;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% Problem 1 on p. 194 of Davis (1962)\r\nth = pi/4;\r\nT_correct = 6.534345229832591;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/3;\r\nT_correct = 6.743001419250384;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/2;\r\nT_correct = 7.416298709205487;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = 36*pi/37;\r\nT_correct = 18.190113206504414;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 72*pi/73;\r\nT_correct = 20.902949604823448;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 0;\r\nT_correct = 2*pi;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('pendulumPeriod.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-06T01:13:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-06-06T00:37:45.000Z","updated_at":"2026-01-09T20:11:24.000Z","published_at":"2022-06-06T01:13:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49830\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 49830\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for the period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"theta0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the initial angle and returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\\sqrt{g/L}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\\\\sqrt{g/L}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"L\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the length of the pendulum and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"g\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":433,"title":"jogging?","description":"Imagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you.  You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.","description_html":"\u003cp\u003eImagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you.  You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.\u003c/p\u003e","function_template":"function degrees = jogging(d,r,s)\r\ndegrees=0;\r\nend","test_suite":"%%\r\nd = 4; r = 1; s = 1; degrees_correct = 60;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 5; r = 1; s = 1; degrees_correct = 66;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 6; r = 1; s = 1; degrees_correct = 71;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 7; r = 1; s = 1; degrees_correct = 73;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 8; r = 1; s = 1; degrees_correct = 76;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 7; r = 1; s = 2; degrees_correct = 55;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":3,"created_by":166,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":"2012-03-02T10:41:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-01T21:44:31.000Z","updated_at":"2026-01-31T12:59:31.000Z","published_at":"2012-03-02T16:52:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eImagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you. You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":52654,"title":"Easy Sequences 13: Average Speed of Spaceship","description":"A certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around. \r\nGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops. Please round-off your answer to the nearest integer.\r\nNOTE: Use clasical physics only. Ignore any relativistic effects.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 165px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eA certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e Please round-off your answer to the nearest integer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eNOTE: Use clasical physics only. Ignore any relativistic effects.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = mean_velocity(s,v)\r\n  y = x;\r\nend","test_suite":"%%\r\ns = 10000;\r\nv = 10000;\r\nv_correct = 1022;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = 1234567;\r\nv = 1234567;\r\nv_correct = 84539;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = '1234567891011121314151617181920';\r\nv = 123456789;\r\nv_correct = 6427156;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = 1e100;\r\nvs = 1:1000;\r\nv_correct = 72076;\r\nassert(isequal(sum(arrayfun(@(v) mean_velocity(s,v),vs)),v_correct))\r\n%%\r\ns = intmax;\r\nv = double(intmax);\r\nv_correct = 97326319;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = intmax('int64')/100;\r\nv = double(intmax('int64'))/100;\r\nv_correct = 2326765408587627;\r\nassert(isequal(mean_velocity(s,v),v_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":4,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-09-05T14:22:29.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-09-05T08:20:36.000Z","updated_at":"2025-12-22T16:16:27.000Z","published_at":"2021-09-05T08:20:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \\\"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Please round-off your answer to the nearest integer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: Use clasical physics only. Ignore any relativistic effects.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60326,"title":"Determine the mass of a bat which strikes a ball","description":"Given the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\r\nRound everything to 2 decimals.\r\nvariables: v, f, s, phi","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 102px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 51px; transform-origin: 407px 51px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eRound everything to 2 decimals.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003evariables: v, f, s, phi\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function m = mass(s,f,v,phi)\r\n  E = m*c^2;\r\nend","test_suite":"%%\r\ns = 2.23;\r\nf = 400\r\nv = 115/3.6\r\nphi = 10\r\ng = 9.81\r\ny_correct = 1.74;\r\nassert(isequal(mass(s,f,v,phi),y_correct))\r\n\r\n%%\r\ns = 2.23;\r\nf = 500\r\nv = 115/3.6\r\nphi = 10\r\ng = 9.81\r\ny_correct = 2.17;\r\nassert(isequal(mass(s,f,v,phi),y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4201666,"edited_by":4201666,"edited_at":"2024-05-16T16:54:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2024-05-16T16:54:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-05-15T21:31:53.000Z","updated_at":"2024-05-16T16:54:48.000Z","published_at":"2024-05-15T21:31:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRound everything to 2 decimals.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003evariables: v, f, s, phi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58778,"title":"Count collisions in an idealized block system","description":"Two blocks, which have masses  and , slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed . All of the collisions are elastic. \r\nWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 358.7px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 179.35px; transform-origin: 407px 179.35px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.525px 8px; transform-origin: 101.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTwo blocks, which have masses \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 231.058px 8px; transform-origin: 231.058px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 14.5px; height: 20px;\" width=\"14.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1833px 8px; transform-origin: 99.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. All of the collisions are elastic. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 371.725px 8px; transform-origin: 371.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 203.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 101.85px; text-align: left; transform-origin: 384px 101.85px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 328px;height: 198px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\" width=\"328\" height=\"198\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = countCollisions(m1,m2,v1)\r\n  y = length(ode45('NewtonII',tspan,y0));\r\n  ","test_suite":"%%\r\nm1 = 1;\r\nm2 = 1;\r\nv1 = 1;\r\ny_correct = 3;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 2;\r\nm2 = 1;\r\nv1 = 1;\r\ny_correct = 5;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 17;\r\nm2 = 4;\r\nv1 = 8;\r\ny_correct = 6;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 15;\r\nm2 = 1.5;\r\nv1 = 0.3;\r\ny_correct = 10;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 3;\r\nm2 = 0.03;\r\nv1 = 0.5;\r\ny_correct = 31;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 4500000;\r\nm2 = 4.5;\r\nv1 = 0.55;\r\ny_correct = 3141;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 1.8e11;\r\nm2 = 18;\r\nv1 = 5.2;\r\ny_correct = 314159;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np2 = regexprep('11.0010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000','\\.','');\r\nfor k = 2:10\r\n    m2 = rand;\r\n    m1 = 4^k*m2;\r\n    v1 = rand;\r\n    y = countCollisions(m1,m2,v1);\r\n    assert(isequal(dec2bin(y),p2(1:k+2)))\r\nend\r\n\r\n%%\r\np7 = regexprep('3.0663651432036134110263402244652226643520650240155443215426431025161154565220002622436103301443233631','\\.','');\r\nfor k = 2:8\r\n    m2 = rand;\r\n    m1 = 49^k*m2;\r\n    v1 = rand;\r\n    y = countCollisions(m1,m2,v1);\r\n    assert(isequal(dec2base(y,7),p7(1:k+1)))\r\nend\r\n\r\n%%\r\nfiletext = fileread('countCollisions.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'classdef'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-07-22T01:52:49.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-22T01:47:49.000Z","updated_at":"2023-07-22T01:52:49.000Z","published_at":"2023-07-22T01:47:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwo blocks, which have masses \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. All of the collisions are elastic. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"198\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"328\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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Center of Mass for a Set of Floating Spheres ","description":"Each sphere has a position determined by theta (x,y plane angle) and tau (elevation angle) as well as L, the distance of the center of the sphere from the origin. Each sphere also has a radius, r, and a density of rho. \r\nThese values are defined in a single input matrix: sceneAttributes\r\nThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\r\nAll angles are in degrees, all distances are in meters, and density is in kg/m^3\r\nAssume density and lengths are always positive","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 162px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 81px; transform-origin: 407px 81px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEach sphere has a position determined by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etheta\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (x,y plane angle) and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etau\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (elevation angle) as well as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eL\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the distance of the center of the sphere from the origin. Each sphere also has a radius, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003er\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and a density of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003erho\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThese values are defined in a single input matrix: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003esceneAttributes\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAll angles are in degrees, all distances are in meters, and density is in kg/m^3\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume density and lengths are always positive\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function pos = sphereCOM(sceneAttributes)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [45, 45, 3, 0.5, 2];\r\ny_correct = [1.5 1.5 2.1213];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\nx = [45, 45, 3, 0.5, 2;\r\n     45, -135, 3, 0.5, 2];\r\ny_correct = [0, 0, 0];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\n%%\r\nx = [45, 45, 3, 0.5, 2;\r\n     135, 45, 3, 0.5, 2;\r\n     225, 45, 3, 0.5, 2;\r\n     315, 45, 3, 0.5, 2];\r\ny_correct = [0, 0, 2.1213];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\nx = [82.0000   70.0000   37.0000    0.3567    5.0000\r\n  -84.0000   69.0000   93.0000    0.5793    2.0000\r\n   11.0000   50.0000   85.0000    0.0155    3.0000\r\n -227.0000   64.0000   18.0000    0.0468    5.0000\r\n -133.0000  -74.0000   10.0000    0.9638    5.0000\r\n -354.0000  -43.0000   86.0000    0.6238    1.0000\r\n   83.0000   32.0000   35.0000    0.9870    5.0000\r\n -293.0000   79.0000    6.0000    0.8137    4.0000\r\n  109.0000   35.0000   62.0000    0.0159    3.0000\r\n -289.0000   62.0000   71.0000    0.7193    1.0000\r\n  105.0000   42.0000    6.0000    0.5844    2.0000\r\n -342.0000    8.0000   38.0000    0.2064    2.0000\r\n   70.0000  -72.0000   33.0000    0.1787    4.0000\r\n -151.0000   35.0000   46.0000    0.5571    1.0000\r\n    3.0000  -70.0000   74.0000    0.5721    5.0000\r\n -177.0000  -12.0000   92.0000    0.2633    4.0000\r\n -103.0000   72.0000    3.0000    0.8341    1.0000\r\n -271.0000  -11.0000    4.0000    0.2545    1.0000\r\n  179.0000   76.0000   97.0000    0.1473    5.0000\r\n -114.0000   -6.0000    4.0000    0.5862    2.0000\r\n   45.0000   73.0000   32.0000    0.0335    5.0000\r\n  337.0000  -42.0000    4.0000    0.9355    5.0000\r\n  121.0000  -84.0000   37.0000    0.9568    4.0000\r\n  144.0000  -11.0000   85.0000    0.5869    3.0000\r\n -241.0000    3.0000    7.0000    0.4570    3.0000\r\n -343.0000  -77.0000   46.0000    0.9695    2.0000\r\n  258.0000  -60.0000   52.0000    0.6969    5.0000\r\n  -47.0000  -74.0000   48.0000    0.2568    3.0000\r\n -212.0000  -26.0000   38.0000    0.4174    5.0000\r\n -335.0000  -12.0000   69.0000    0.0045    3.0000\r\n  -33.0000  -66.0000   48.0000    0.5825    3.0000\r\n  161.0000  -85.0000   99.0000    0.4631    4.0000\r\n  218.0000   -7.0000   34.0000    0.4395    5.0000\r\n   57.0000  -74.0000   99.0000    0.8935    1.0000\r\n   -8.0000  -48.0000   66.0000    0.7985    3.0000\r\n  -98.0000   30.0000   69.0000    0.2574    2.0000\r\n  173.0000   23.0000   86.0000    0.8359    2.0000\r\n  186.0000  -49.0000   52.0000    0.0849    2.0000\r\n  -68.0000  -50.0000    9.0000    0.0143    4.0000\r\n  202.0000  -83.0000   88.0000    0.7898    3.0000\r\n -179.0000  -79.0000   11.0000    0.5538    5.0000\r\n  -76.0000   10.0000   67.0000    0.8318    4.0000\r\n  266.0000  -83.0000   49.0000    0.0301    4.0000\r\n -348.0000   80.0000   70.0000    0.0598    2.0000\r\n  309.0000   49.0000   62.0000    0.4251    1.0000\r\n  296.0000   -6.0000   20.0000    0.3192    5.0000\r\n  266.0000   77.0000   59.0000    0.8861    5.0000\r\n  144.0000  -77.0000   12.0000    0.6542    4.0000\r\n  -64.0000   54.0000   21.0000    0.0737    5.0000\r\n   71.0000  -79.0000   93.0000    0.9238    5.0000\r\n -170.0000  -34.0000   29.0000    0.5234    4.0000\r\n  111.0000   32.0000   88.0000    0.8344    3.0000\r\n   11.0000  -59.0000  100.0000    0.0446    1.0000\r\n  -35.0000   70.0000   71.0000    0.0496    3.0000\r\n  207.0000   52.0000   10.0000    0.5526    1.0000\r\n  209.0000  -89.0000   89.0000    0.5076    5.0000\r\n  309.0000   -2.0000   42.0000    0.1053    5.0000\r\n  353.0000  -62.0000   58.0000    0.6336    3.0000\r\n -191.0000  -59.0000   29.0000    0.1629    4.0000\r\n -262.0000         0   34.0000    0.5343    4.0000\r\n   28.0000  -47.0000   98.0000    0.7263    5.0000\r\n  328.0000   21.0000   43.0000    0.1479    3.0000\r\n -341.0000    8.0000   91.0000    0.5571    3.0000\r\n -324.0000  -60.0000   92.0000    0.1671    4.0000\r\n  -12.0000  -24.0000   83.0000    0.8690    2.0000\r\n   91.0000   30.0000   61.0000    0.4496    3.0000\r\n  230.0000  -20.0000   16.0000    0.3626    1.0000\r\n  128.0000   65.0000   82.0000    0.0067    1.0000\r\n   14.0000  -89.0000   61.0000    0.1293    4.0000\r\n  330.0000   39.0000   55.0000    0.0746    1.0000\r\n   71.0000   25.0000   27.0000    0.4566    2.0000\r\n  -32.0000   50.0000   74.0000    0.3240    1.0000\r\n  118.0000  -50.0000   54.0000    0.6298    5.0000\r\n  -55.0000   71.0000   46.0000    0.3064    3.0000\r\n -297.0000   47.0000   51.0000    0.8419    5.0000\r\n -114.0000   62.0000   42.0000    0.3906    3.0000\r\n   31.0000   69.0000   16.0000    0.1525    4.0000\r\n -284.0000   61.0000   18.0000    0.9337    4.0000\r\n  287.0000  -56.0000   38.0000    0.7202    5.0000\r\n  -43.0000   57.0000   11.0000    0.2651    4.0000\r\n -108.0000  -34.0000   99.0000    0.3242    4.0000\r\n   47.0000   -9.0000   21.0000    0.4719    1.0000\r\n  241.0000  -66.0000   36.0000    0.9853    4.0000\r\n  107.0000   55.0000   75.0000    0.5283    5.0000\r\n  -21.0000  -30.0000   69.0000    0.7108    3.0000\r\n -154.0000    8.0000   13.0000    0.9318    1.0000\r\n -182.0000   88.0000    1.0000    0.6157    2.0000\r\n  106.0000  -58.0000   23.0000    0.1413    5.0000\r\n  195.0000  -77.0000   52.0000    0.2124    5.0000\r\n -180.0000   50.0000   14.0000    0.8675    3.0000\r\n  -43.0000   36.0000   17.0000    0.2018    1.0000\r\n -147.0000  -80.0000   30.0000    0.1032    5.0000\r\n  -99.0000   36.0000   22.0000    0.4703    5.0000\r\n  198.0000    5.0000   24.0000    0.2535    4.0000\r\n   33.0000   -9.0000  100.0000    0.1854    5.0000\r\n  234.0000   18.0000   49.0000    0.3313    3.0000\r\n  200.0000   85.0000   26.0000    0.1587    5.0000\r\n  305.0000   75.0000  100.0000    0.4419    2.0000\r\n   36.0000   87.0000   67.0000    0.1831    5.0000\r\n  -50.0000   86.0000    7.0000    0.7323    4.0000];\r\ny_correct = [3.6203    3.4188  -11.9005];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\n%Cheating is bad\r\nfunctions={'!','feval','eval','str2func','str2num','regex','system','dos','unix','perl','assert','fopen','write','save','setenv','path','please'};\r\nassessFunctionAbsence(functions, 'FileName', 'sphereCOM.m');","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":2002950,"edited_by":2002950,"edited_at":"2022-09-07T15:51:11.000Z","deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":"2022-09-07T15:51:11.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-03-03T14:59:29.000Z","updated_at":"2022-09-07T15:51:49.000Z","published_at":"2022-03-03T14:59:29.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach sphere has a position determined by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etheta\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (x,y plane angle) and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etau\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (elevation angle) as well as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the distance of the center of the sphere from the origin. Each sphere also has a radius, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a density of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003erho\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThese values are defined in a single input matrix: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esceneAttributes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll angles are in degrees, all distances are in meters, and density is in kg/m^3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume density and lengths are always positive\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49652,"title":"Find the spot diameter from the intensity distribution matrix of single spot (circle)","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 445.903px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.493px 222.951px; transform-origin: 406.493px 222.951px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIntensity distribution will be same as \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Normal_distribution\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egaussian dsitribution\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and check for \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egaussian beam distribution\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 40px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 20px; text-align: left; transform-origin: 383.498px 20px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDistance between each index will be equal to 1 mm. Grid will be n*n matrix of distribution . n will be odd number . If n=3 center of the spot will be (2,2)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind spot diameter in mm.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eHint : w(z) is the radius of the laser beam where the irradiance is 1/e2 (13.5%) of Intensity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eexample \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ex    0.0000    0.0045    0.0335    0.0045    0.0000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0335   13.5335  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e100.0000 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  13.5335    0.0335\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0000    0.0045    0.0335    0.0045    0.0000]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI=100 (max intensity);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI(13.5 %)=13.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eradius=1 mm;Spot diameter=2 mm\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Int(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('Int.m');\r\nassert(isempty(strfind(filetext, 'regexp')),'regexp hacks are forbidden')\r\nassert(isempty(strfind(filetext, 'if')),'if is forbidden')\r\nassert(isempty(strfind(filetext, 'str')),'str is forbidden')\r\nassert(isempty(strfind(filetext, 'while')),'while is forbidden')\r\nassert(isempty(strfind(filetext, 'switch')),'switch is forbidden')\r\n\r\n\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0002    0.0045    0.0123    0.0045    0.0002    0.0000    0.0000\r\n    0.0000    0.0002    0.0335    0.6738    1.8316    0.6738    0.0335    0.0002    0.0000\r\n    0.0000    0.0045    0.6738   13.5335   36.7879   13.5335    0.6738    0.0045    0.0000\r\n    0.0000    0.0123    1.8316   36.7879  100.0000   36.7879    1.8316    0.0123    0.0000\r\n    0.0000    0.0045    0.6738   13.5335   36.7879   13.5335    0.6738    0.0045    0.0000\r\n    0.0000    0.0002    0.0335    0.6738    1.8316    0.6738    0.0335    0.0002    0.0000\r\n    0.0000    0.0000    0.0002    0.0045    0.0123    0.0045    0.0002    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),4))\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0003    0.0069    0.0189    0.0069    0.0003    0.0000    0.0000\r\n    0.0000    0.0003    0.0513    1.0309    2.8023    1.0309    0.0513    0.0003    0.0000\r\n    0.0000    0.0069    1.0309   20.7063   56.2856   20.7063    1.0309    0.0069    0.0000\r\n    0.0000    0.0189    2.8023   56.2856  153.0000   56.2856    2.8023    0.0189    0.0000\r\n    0.0000    0.0069    1.0309   20.7063   56.2856   20.7063    1.0309    0.0069    0.0000\r\n    0.0000    0.0003    0.0513    1.0309    2.8023    1.0309    0.0513    0.0003    0.0000\r\n    0.0000    0.0000    0.0003    0.0069    0.0189    0.0069    0.0003    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),4))\r\n\r\n%%\r\nx=[ 0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0003    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0188    0.1387    0.0188    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0003    0.1387    1.0250    0.1387    0.0003    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0188    0.1387    0.0188    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0003    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),2))\r\n\r\n%%\r\nx = [    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0005    0.0007    0.0005    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0091    0.0407    0.0671    0.0407    0.0091    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0247    0.3007    1.3476    2.2218    1.3476    0.3007    0.0247    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0091    0.3007    3.6631   16.4170   27.0671   16.4170    3.6631    0.3007    0.0091    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0005    0.0407    1.3476   16.4170   73.5759  121.3061   73.5759   16.4170    1.3476    0.0407    0.0005    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0007    0.0671    2.2218   27.0671  121.3061  200.0000  121.3061   27.0671    2.2218    0.0671    0.0007    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0005    0.0407    1.3476   16.4170   73.5759  121.3061   73.5759   16.4170    1.3476    0.0407    0.0005    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0091    0.3007    3.6631   16.4170   27.0671   16.4170    3.6631    0.3007    0.0091    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0247    0.3007    1.3476    2.2218    1.3476    0.3007    0.0247    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0091    0.0407    0.0671    0.0407    0.0091    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0005    0.0007    0.0005    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),8))\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0036    0.0070    0.0036    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0018    0.0517    0.3818    0.7436    0.3818    0.0517    0.0018    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0517    1.4484   10.7022   20.8450   10.7022    1.4484    0.0517    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0036    0.3818   10.7022   79.0791  154.0251   79.0791   10.7022    0.3818    0.0036    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0070    0.7436   20.8450  154.0251  300.0000  154.0251   20.8450    0.7436    0.0070    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0036    0.3818   10.7022   79.0791  154.0251   79.0791   10.7022    0.3818    0.0036    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0517    1.4484   10.7022   20.8450   10.7022    1.4484    0.0517    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0018    0.0517    0.3818    0.7436    0.3818    0.0517    0.0018    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0036    0.0070    0.0036    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000]\r\n\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[   0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0002    0.0018    0.0035    0.0018    0.0002    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0009    0.0262    0.1934    0.3768    0.1934    0.0262    0.0009    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0002    0.0262    0.7338    5.4224   10.5615    5.4224    0.7338    0.0262    0.0002    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0018    0.1934    5.4224   40.0668   78.0394   40.0668    5.4224    0.1934    0.0018    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0035    0.3768   10.5615   78.0394  152.0000   78.0394   10.5615    0.3768    0.0035    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0018    0.1934    5.4224   40.0668   78.0394   40.0668    5.4224    0.1934    0.0018    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0002    0.0262    0.7338    5.4224   10.5615    5.4224    0.7338    0.0262    0.0002    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0009    0.0262    0.1934    0.3768    0.1934    0.0262    0.0009    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0002    0.0018    0.0035    0.0018    0.0002    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[ 0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0016    0.0120    0.0233    0.0120    0.0016    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0001    0.0061    0.1721    1.2714    2.4763    1.2714    0.1721    0.0061    0.0001    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0016    0.1721    4.8231   35.6383   69.4140   35.6383    4.8231    0.1721    0.0016    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0120    1.2714   35.6383  263.3335  512.9037  263.3335   35.6383    1.2714    0.0120    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0001    0.0233    2.4763   69.4140  512.9037  999.0000  512.9037   69.4140    2.4763    0.0233    0.0001    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0120    1.2714   35.6383  263.3335  512.9037  263.3335   35.6383    1.2714    0.0120    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0016    0.1721    4.8231   35.6383   69.4140   35.6383    4.8231    0.1721    0.0016    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0001    0.0061    0.1721    1.2714    2.4763    1.2714    0.1721    0.0061    0.0001    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0016    0.0120    0.0233    0.0120    0.0016    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[    0.0001    0.0007    0.0044    0.0197    0.0633    0.1456    0.2401    0.2837    0.2401    0.1456    0.0633    0.0197    0.0044    0.0007    0.0001\r\n    0.0007    0.0061    0.0384    0.1721    0.5525    1.2714    2.0961    2.4763    2.0961    1.2714    0.5525    0.1721    0.0384    0.0061    0.0007\r\n    0.0044    0.0384    0.2401    1.0762    3.4559    7.9520   13.1106   15.4883   13.1106    7.9520    3.4559    1.0762    0.2401    0.0384    0.0044\r\n    0.0197    0.1721    1.0762    4.8231   15.4883   35.6383   58.7577   69.4140   58.7577   35.6383   15.4883    4.8231    1.0762    0.1721    0.0197\r\n    0.0633    0.5525    3.4559   15.4883   49.7373  114.4443  188.6867  222.9070  188.6867  114.4443   49.7373   15.4883    3.4559    0.5525    0.0633\r\n    0.1456    1.2714    7.9520   35.6383  114.4443  263.3335  434.1636  512.9037  434.1636  263.3335  114.4443   35.6383    7.9520    1.2714    0.1456\r\n    0.2401    2.0961   13.1106   58.7577  188.6867  434.1636  715.8148  845.6352  715.8148  434.1636  188.6867   58.7577   13.1106    2.0961    0.2401\r\n    0.2837    2.4763   15.4883   69.4140  222.9070  512.9037  845.6352  999.0000  845.6352  512.9037  222.9070   69.4140   15.4883    2.4763    0.2837\r\n    0.2401    2.0961   13.1106   58.7577  188.6867  434.1636  715.8148  845.6352  715.8148  434.1636  188.6867   58.7577   13.1106    2.0961    0.2401\r\n    0.1456    1.2714    7.9520   35.6383  114.4443  263.3335  434.1636  512.9037  434.1636  263.3335  114.4443   35.6383    7.9520    1.2714    0.1456\r\n    0.0633    0.5525    3.4559   15.4883   49.7373  114.4443  188.6867  222.9070  188.6867  114.4443   49.7373   15.4883    3.4559    0.5525    0.0633\r\n    0.0197    0.1721    1.0762    4.8231   15.4883   35.6383   58.7577   69.4140   58.7577   35.6383   15.4883    4.8231    1.0762    0.1721    0.0197\r\n    0.0044    0.0384    0.2401    1.0762    3.4559    7.9520   13.1106   15.4883   13.1106    7.9520    3.4559    1.0762    0.2401    0.0384    0.0044\r\n    0.0007    0.0061    0.0384    0.1721    0.5525    1.2714    2.0961    2.4763    2.0961    1.2714    0.5525    0.1721    0.0384    0.0061    0.0007\r\n    0.0001    0.0007    0.0044    0.0197    0.0633    0.1456    0.2401    0.2837    0.2401    0.1456    0.0633    0.0197    0.0044    0.0007    0.0001];\r\n\r\nassert(isequal(Int(x),24))\r\n\r\n%%\r\nx=[  0.0000    0.0001    0.0005    0.0023    0.0077    0.0208    0.0454    0.0790    0.1103    0.1233    0.1103    0.0790    0.0454    0.0208    0.0077    0.0023    0.0005    0.0001    0.0000\r\n    0.0001    0.0007    0.0035    0.0149    0.0507    0.1378    0.2999    0.5227    0.7294    0.8152    0.7294    0.5227    0.2999    0.1378    0.0507    0.0149    0.0035    0.0007    0.0001\r\n    0.0005    0.0035    0.0186    0.0790    0.2683    0.7294    1.5877    2.7673    3.8621    4.3159    3.8621    2.7673    1.5877    0.7294    0.2683    0.0790    0.0186    0.0035    0.0005\r\n    0.0023    0.0149    0.0790    0.3351    1.1377    3.0925    6.7312   11.7319   16.3732   18.2973   16.3732   11.7319    6.7312    3.0925    1.1377    0.3351    0.0790    0.0149    0.0023\r\n    0.0077    0.0507    0.2683    1.1377    3.8621   10.4982   22.8506   39.8265   55.5824   62.1143   55.5824   39.8265   22.8506   10.4982    3.8621    1.1377    0.2683    0.0507    0.0077\r\n    0.0208    0.1378    0.7294    3.0925   10.4982   28.5369   62.1143  108.2597  151.0885  168.8443  151.0885  108.2597   62.1143   28.5369   10.4982    3.0925    0.7294    0.1378    0.0208\r\n    0.0454    0.2999    1.5877    6.7312   22.8506   62.1143  135.1999  235.6412  328.8638  367.5116  328.8638  235.6412  135.1999   62.1143   22.8506    6.7312    1.5877    0.2999    0.0454\r\n    0.0790    0.5227    2.7673   11.7319   39.8265  108.2597  235.6412  410.7012  573.1797  640.5392  573.1797  410.7012  235.6412  108.2597   39.8265   11.7319    2.7673    0.5227    0.0790\r\n    0.1103    0.7294    3.8621   16.3732   55.5824  151.0885  328.8638  573.1797  799.9367  893.9445  799.9367  573.1797  328.8638  151.0885   55.5824   16.3732    3.8621    0.7294    0.1103\r\n    0.1233    0.8152    4.3159   18.2973   62.1143  168.8443  367.5116  640.5392  893.9445  999.0000  893.9445  640.5392  367.5116  168.8443   62.1143   18.2973    4.3159    0.8152    0.1233\r\n    0.1103    0.7294    3.8621   16.3732   55.5824  151.0885  328.8638  573.1797  799.9367  893.9445  799.9367  573.1797  328.8638  151.0885   55.5824   16.3732    3.8621    0.7294    0.1103\r\n    0.0790    0.5227    2.7673   11.7319   39.8265  108.2597  235.6412  410.7012  573.1797  640.5392  573.1797  410.7012  235.6412  108.2597   39.8265   11.7319    2.7673    0.5227    0.0790\r\n    0.0454    0.2999    1.5877    6.7312   22.8506   62.1143  135.1999  235.6412  328.8638  367.5116  328.8638  235.6412  135.1999   62.1143   22.8506    6.7312    1.5877    0.2999    0.0454\r\n    0.0208    0.1378    0.7294    3.0925   10.4982   28.5369   62.1143  108.2597  151.0885  168.8443  151.0885  108.2597   62.1143   28.5369   10.4982    3.0925    0.7294    0.1378    0.0208\r\n    0.0077    0.0507    0.2683    1.1377    3.8621   10.4982   22.8506   39.8265   55.5824   62.1143   55.5824   39.8265   22.8506   10.4982    3.8621    1.1377    0.2683    0.0507    0.0077\r\n    0.0023    0.0149    0.0790    0.3351    1.1377    3.0925    6.7312   11.7319   16.3732   18.2973   16.3732   11.7319    6.7312    3.0925    1.1377    0.3351    0.0790    0.0149    0.0023\r\n    0.0005    0.0035    0.0186    0.0790    0.2683    0.7294    1.5877    2.7673    3.8621    4.3159    3.8621    2.7673    1.5877    0.7294    0.2683    0.0790    0.0186    0.0035    0.0005\r\n    0.0001    0.0007    0.0035    0.0149    0.0507    0.1378    0.2999    0.5227    0.7294    0.8152    0.7294    0.5227    0.2999    0.1378    0.0507    0.0149    0.0035    0.0007    0.0001\r\n    0.0000    0.0001    0.0005    0.0023    0.0077    0.0208    0.0454    0.0790    0.1103    0.1233    0.1103    0.0790    0.0454    0.0208    0.0077    0.0023    0.0005    0.0001    0.0000];\r\n\r\nassert(isequal(Int(x),36))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":610936,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2021-01-05T18:44:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-28T20:35:54.000Z","updated_at":"2025-12-09T19:47:07.000Z","published_at":"2020-12-28T21:31:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIntensity distribution will be same as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Normal_distribution\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egaussian dsitribution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and check for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egaussian beam distribution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDistance between each index will be equal to 1 mm. Grid will be n*n matrix of distribution . n will be odd number . If n=3 center of the spot will be (2,2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind spot diameter in mm.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHint : w(z) is the radius of the laser beam where the irradiance is 1/e2 (13.5%) of Intensity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eexample \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex    0.0000    0.0045    0.0335    0.0045    0.0000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0335   13.5335  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e100.0000 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  13.5335    0.0335\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0000    0.0045    0.0335    0.0045    0.0000]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI=100 (max intensity);\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI(13.5 %)=13.5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eradius=1 mm;Spot diameter=2 mm\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44741,"title":"You are constantly moving at a speed v faster than your twin brother. How long does it take before you become 1s younger than him according to the theory of relativity?","description":"You are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\r\n\r\nDefine:\r\ngam = 1/sqrt(1-v^2/c^2)   where v is the relative speed in m/s and c is the speed of light in m/s.\r\n\r\nAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?","description_html":"\u003cp\u003eYou are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\u003c/p\u003e\u003cp\u003eDefine:\r\ngam = 1/sqrt(1-v^2/c^2)   where v is the relative speed in m/s and c is the speed of light in m/s.\u003c/p\u003e\u003cp\u003eAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?\u003c/p\u003e","function_template":"function years = becomeOneSecondYounger(v_in)\r\n    \r\nend","test_suite":"%%\r\nv_in = 100;            % km/h\r\nyearsCorrect = 241830; % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c10)\r\n\r\n\r\n%%\r\nv_in = 1000;           % km/h\r\nyearsCorrect = 2418.7; % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c0.1)\r\n\r\n%%\r\nv_in = 10000;           % km/h\r\nyearsCorrect = 24.187;  % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c0.01)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":195293,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2018-10-03T10:36:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-03T08:07:37.000Z","updated_at":"2018-10-03T10:36:13.000Z","published_at":"2018-10-03T10:32:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefine: gam = 1/sqrt(1-v^2/c^2) where v is the relative speed in m/s and c is the speed of light in m/s.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2316,"title":"Spin Matrices","description":"The spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\r\nSuppose you have integer or half-integer spin of value s. The matrices Sx, Sy and Sz representing it have the following properties:\r\nSi (with i={x,y,z}) are traceless Hermitian matrices;\r\nCommutation relations (a): [ Si,Sj ] = i εijk Sk, where [·,·] is the commutator and εijk is the Levi-Civita symbol.\r\nCommutation relations (b): [ Si,S² ] = 0, where S² = Sx²+Sy²+Sz²;\r\nEigenvalues: S² = j(j+1)·I and Sz = diag( -j/2, -j/2+1, … ,j/2-1, j ), where I is the identity matrix.\r\nSee also this article for more reference.\r\nExamples\r\n [Sx,Sy,Sz] = spin_matrices(1/2)\r\n\r\n Sx = \r\n     0      0.5\r\n     0.5    0\r\n\r\n Sy = \r\n     0     -0.5i\r\n     0.5i   0\r\n\r\n Sz = \r\n     0.5    0\r\n     0     -0.5\r\nNote:\r\nThe usual cheats are not allowed!","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 592.367px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 296.183px; transform-origin: 407px 296.183px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 171.5px 8px; transform-origin: 171.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose you have integer or half-integer spin of value\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003es\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45px 8px; transform-origin: 45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The matrices\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.5px 8px; transform-origin: 7.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSx\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSy\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSz\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108px 8px; transform-origin: 108px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e representing it have the following properties:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 40.8667px; transform-origin: 391px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.5px 8px; transform-origin: 16.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (with\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei={x,y,z}\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15px 8px; transform-origin: 15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etraceless\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eHermitian matrices\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 87.5px 8px; transform-origin: 87.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCommutation relations (a): [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi,Sj\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.5px 8px; transform-origin: 13.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ] = i\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eεijk Sk\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 55px 8px; transform-origin: 55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where [·,·] is the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ecommutator\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eεijk\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19px 8px; transform-origin: 19px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eLevi-Civita symbol\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 87.5px 8px; transform-origin: 87.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCommutation relations (b): [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15px 8px; transform-origin: 15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi,S²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39px 8px; transform-origin: 39px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ] = 0, where\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eS²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e =\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39px 8px; transform-origin: 39px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSx²+Sy²+Sz²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.5px 8px; transform-origin: 40.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEigenvalues:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eS²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e =\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej(j+1)·I\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSz\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = diag(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.5px 8px; transform-origin: 66.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e-j/2, -j/2+1, … ,j/2-1, j\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 27.5px 8px; transform-origin: 27.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ), where\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eI\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66px 8px; transform-origin: 66px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the identity matrix.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.5px 8px; transform-origin: 27.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ethis article\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63px 8px; transform-origin: 63px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for more reference.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 33.5px 8px; transform-origin: 33.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 265.633px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 132.817px; transform-origin: 404px 132.817px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 128px 8.5px; tab-size: 4; transform-origin: 128px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e [Sx,Sy,Sz] = spin_matrices(1/2)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sx = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0      0.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5    0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sy = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 64px 8.5px; tab-size: 4; transform-origin: 64px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     -0.5i\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5i   0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sz = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5    0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     -0.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.5px 8px; transform-origin: 18.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNote:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.5px 8px; transform-origin: 54.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe usual cheats\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eare not\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e allowed!\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [Sx,Sy,Sz] = spin_matrices(s)\r\n  [Sx,Sy,Sz] = deal(s);\r\nend","test_suite":"%%\r\nuser_solution = fileread('spin_matrices.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'num2str')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'fprintf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\n% use auxiliary functions\r\niseq = @(x,y)norm(x-y)\u003c=64*eps; % check if equal up to 64 eps\r\ncom  = @(x,y)x*y-y*x;           % commutator\r\ntrace= @(x)  sum(diag(x));      % trace\r\n%\r\nfprintf('Testing...\\n')\r\nfor s = 1/2:1/2:5,\r\n   % Get the matrices\r\n   fprintf('\\ts=%-3s : ',strtrim(rats(s)));\r\n   [Sx,Sy,Sz] = spin_matrices(s);\r\n   %\r\n   % ancillary parameters\r\n   mz   = (-s:s)';                            % eigenvalues\r\n   S2   = Sx^2+Sy^2+Sz^2;                     % S^2 matrix\r\n   %\r\n   assert(trace(Sx)==0\u0026\u0026iseq(Sx,Sx'),'Sx must be a traceless Hermitian matrix');\r\n   assert(trace(Sy)==0\u0026\u0026iseq(Sy,Sy'),'Sy must be a traceless Hermitian matrix');\r\n   assert(trace(Sz)==0\u0026\u0026iseq(Sz,Sz'),'Sz must be a traceless Hermitian matrix');\r\n   %\r\n   % actual values\r\n   assert(iseq(com(Sx,Sy),1i*Sz), 'Commutation relations: [Sx,Sy] = i*Sz')\r\n   assert(iseq(com(Sy,Sz),1i*Sx), 'Commutation relations: [Sy,Sz] = i*Sx')\r\n   assert(iseq(com(Sz,Sx),1i*Sy), 'Commutation relations: [Sz,Sx] = i*Sy')\r\n   %\r\n   assert(iseq(S2,s*(s+1)*eye(2*s+1)), 'S^2 must be a quantum number!');\r\n   assert(iseq(eig(Sz),mz),            'Sz must be a quantum number!');\r\n   %\r\n   fprintf('OK!\\n');\r\nend\r\n%\r\nfprintf('\\n \\nWolfgang Pauli would be proud!\\n')\r\n%","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":10352,"edited_by":223089,"edited_at":"2022-08-20T10:34:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":"2014-05-11T14:57:22.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-05-11T11:28:29.000Z","updated_at":"2024-11-18T01:26:43.000Z","published_at":"2014-05-11T14:54:32.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you have integer or half-integer spin of value\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The matrices\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSx\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSy\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSz\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e representing it have the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (with\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei={x,y,z}\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etraceless\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHermitian matrices\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCommutation relations (a): [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi,Sj\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ] = i\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eεijk Sk\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, where [·,·] is the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ecommutator\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eεijk\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eLevi-Civita symbol\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCommutation relations (b): [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi,S²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ] = 0, where\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e =\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSx²+Sy²+Sz²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEigenvalues:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e =\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej(j+1)·I\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSz\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = diag(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-j/2, -j/2+1, … ,j/2-1, j\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), where\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the identity matrix.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ethis article\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for more reference.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ [Sx,Sy,Sz] = spin_matrices(1/2)\\n\\n Sx = \\n     0      0.5\\n     0.5    0\\n\\n Sy = \\n     0     -0.5i\\n     0.5i   0\\n\\n Sz = \\n     0.5    0\\n     0     -0.5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNote:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe usual cheats\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare not\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57770,"title":"Number of images formed due to two inclined mirrors and their coordinates","description":"There are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. Final output should contain a vector of coordinates including the original object coordinate.\r\n ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 249px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 124.5px; transform-origin: 407px 124.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThere are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. Final output should contain a vector of coordinates including the original object coordinate.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 156px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 78px; text-align: left; transform-origin: 384px 78px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"170\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [N,y] = images(x,theta)\r\n  \r\nend","test_suite":"%%\r\nx = 2;\r\ntheta = pi;\r\nN_correct = 1;\r\ny_correct = [2 -2;0 0];\r\n[N,y] = images(x,theta)\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n\r\n%%\r\nx = 20;\r\ntheta = pi/4;\r\nN_correct = 7;\r\ny_correct =   [20    14     0   -14   -20   -14     0    14;\r\n                0    14    20    14     0   -14   -20   -14]\r\n\r\n[N,y] = images(x,theta)\r\n    \r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%% \r\nx = 1.1\r\ntheta = pi/2;\r\nN_correct = 3;\r\ny_correct =   [1.1000         0   -1.0000         0;\r\n                    0    1.0000         0   -1.0000];\r\n[N,y] = images(x,theta)\r\n\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%%\r\nx = -10;\r\ntheta = 2*pi;\r\nN_correct = 0;\r\ny_correct = [-10;0];\r\n[N,y] = images(x,theta)\r\n\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%%\r\nx = 20;\r\ntheta = 0;\r\nN_correct = Inf;\r\n[N,y] = images(x,theta);\r\n\r\nassert(isequal(N,N_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":1809490,"edited_by":1809490,"edited_at":"2023-03-12T09:24:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2023-03-12T09:24:24.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-03-12T08:45:13.000Z","updated_at":"2023-03-12T09:24:24.000Z","published_at":"2023-03-12T09:10:51.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. 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Partial Differential Equations: Explicit Method","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 294px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 147px; transform-origin: 407px 147px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI got this example from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003enumericalmethodsguy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  youtube video. He solves the problem step by step.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=p0V1eSlM2xo\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOn the second test: I varied the length of rod and the final time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efrom t = 0 and t = 102 seconds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pderod(L,tf)\r\n% defined constants\r\nk=54;                %W/(m*K)\r\np=7800;              %kg/m^3\r\nC=490;               %J/(kg*K)\r\nL0 = 0;               %meter \r\nnode = 5;            % segments\r\ndt = 3;              %second time steps\r\nt0=0;                %s initial time\r\nT0 = 100;           %C initial Temperature   \r\nTend = 25;          %C final Temperature\r\nTm1 =20;            %C\r\nTm = repelem(Tm1,node-1)'; %C middle Temperature \r\nTi=[T0;Tm;Tend];     % Celsius at Time 0second \r\nT0=Ti;               % Celsius at Time 0second \r\n\r\nend","test_suite":"%%\r\n% INPUTS\r\nL = .05;                     %meter\r\ntf = 9;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   53.9089   59.0725   65.9508   85.0000\r\n   20.0000   20.0000   34.3727   39.1307   70.0000\r\n   20.0000   20.0000   20.8983   27.2639   55.0000\r\n   20.0000   22.1193   22.4420   22.8719   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend\r\n%%\r\n% INPUTS\r\nL = .10;                     %meter\r\ntf = 102;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   28.4772   35.1579   40.5179   44.8926   48.5211   51.5762   54.1838   56.4375   58.4071   60.1460   61.6949   63.0856   64.3431   65.4874   66.5344   67.4973   68.3867   69.2115   69.9791   70.6957   71.3666   71.9963   72.5887   73.1472   73.6746   74.1735   74.6463   75.0947   75.5206   75.9256   76.3111   76.6782   77.0283   77.3622   85.0000\r\n   20.0000   20.0000   20.8983   22.3201   24.0280   25.8726   27.7607   29.6354   31.4629   33.2239   34.9089   36.5138   38.0384   39.4848   40.8559   42.1556   43.3879   44.5569   45.6666   46.7206   47.7226   48.6758   49.5832   50.4477   51.2717   52.0578   52.8081   53.5245   54.2090   54.8632   55.4887   56.0871   56.6596   57.2077   57.7323   70.0000\r\n   20.0000   20.0000   20.0561   20.2398   20.5707   21.0425   21.6372   22.3328   23.1075   23.9414   24.8173   25.7209   26.6402   27.5657   28.4898   29.4063   30.3106   31.1990   32.0688   32.9179   33.7447   34.5484   35.3281   36.0835   36.8145   37.5211   38.2035   38.8621   39.4971   40.1091   40.6987   41.2663   41.8126   42.3382   42.8437   55.0000\r\n   20.0000   20.5298   20.9474   21.2824   21.5658   21.8243   22.0780   22.3409   22.6218   22.9253   23.2528   23.6037   23.9760   24.3669   24.7729   25.1908   25.6173   26.0492   26.4837   26.9183   27.3508   27.7792   28.2020   28.6178   29.0256   29.4244   29.8135   30.1925   30.5609   30.9186   31.2653   31.6010   31.9257   32.2395   32.5425   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":227209,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-31T20:44:36.000Z","updated_at":"2020-08-17T00:46:40.000Z","published_at":"2020-07-31T21:36:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI got this example from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\\\"\u003e\u003cw:r\u003e\u003cw:t\u003enumericalmethodsguy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e  youtube video. He solves the problem step by step.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=p0V1eSlM2xo\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the second test: I varied the length of rod and the final time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efrom t = 0 and t = 102 seconds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1109,"title":"USC Spring 2012 ACM: Armageddon","description":"This Challenge is to solve Question E, Armageddon, of the \u003chttp://contest.usc.edu/index.php/Spring12/Home USC ACM Spring 2012 Contest\u003e.\r\n\r\nAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u003e0, \u003c180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km\r\n, and angle in degrees.\r\n\r\nReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places. \r\n\r\n\r\n*Input: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]*\r\n\r\n*Output: [Maximum Time Prior to Launch]*; if too late return -1;\r\n\r\n\r\nThe full \u003chttp://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=armageddon.in.txt USC data file\u003e\r\n\r\n*Example:*\r\n\r\n*Input: 63781.0 20.9514 6378.1 6378.1*\r\n\r\n*Output: 0.00* as immediate Launch is required  \r\n\r\nInput 47835.75,15,6000,5000 returns -1.\r\n\r\n\r\n\u003chttp://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=numerology_darryl.cpp.txt The Judges' E solution\u003e.\r\n\r\nGeometry Hint: Draw a circle and a line from the center of  approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\r\n","description_html":"\u003cp\u003eThis Challenge is to solve Question E, Armageddon, of the \u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home\"\u003eUSC ACM Spring 2012 Contest\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u003e0, \u0026lt;180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km\r\n, and angle in degrees.\u003c/p\u003e\u003cp\u003eReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput: [Maximum Time Prior to Launch]\u003c/b\u003e; if too late return -1;\u003c/p\u003e\u003cp\u003eThe full \u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=armageddon.in.txt\"\u003eUSC data file\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput: 63781.0 20.9514 6378.1 6378.1\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput: 0.00\u003c/b\u003e as immediate Launch is required\u003c/p\u003e\u003cp\u003eInput 47835.75,15,6000,5000 returns -1.\u003c/p\u003e\u003cp\u003e\u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=numerology_darryl.cpp.txt\"\u003eThe Judges' E solution\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eGeometry Hint: Draw a circle and a line from the center of  approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\u003c/p\u003e","function_template":"function [t_launch]=Armageddon(xa,angle,va,vm)\r\n  t_launch=-1;\r\nend","test_suite":"%%\r\n% Armegeddon\r\ntic\r\nurlwrite('http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=armageddon.in.txt','armageddon.in.txt')\r\ntoc\r\n%%\r\n fid=fopen('armageddon.in.txt','r');\r\n \r\n t_expect=[0.00 -1 20.55 -1 -1 28.38 -1 11.03 2.62 4.22 13.15 9.94 61.33 13.56 -1];\r\n \r\n  \r\n qty=fscanf(fid,'%i',1);\r\n for q=1:qty %qty\r\n  n = fscanf(fid,'%f %f %f %f \\n',4)'; % dist, angle, vel A, vel Missile\r\n  xa=n(1);\r\n  angle=n(2);\r\n  va=n(3);\r\n  vm=n(4);\r\n  \r\n  [t]=Armageddon(xa,angle,va,vm) ;\r\n  \r\n  \r\n  assert(isequal(t,t_expect(q)))\r\n  \r\n   \r\n  end\r\n   \r\n fclose(fid);\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":"2012-12-09T06:02:07.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-12-08T21:43:00.000Z","updated_at":"2012-12-09T15:08:14.000Z","published_at":"2012-12-08T22:02:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to solve Question E, Armageddon, of the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eUSC ACM Spring 2012 Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u0026gt;0, \u0026lt;180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km , and angle in degrees.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput: [Maximum Time Prior to Launch]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; if too late return -1;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe full\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=armageddon.in.txt\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eUSC data file\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput: 63781.0 20.9514 6378.1 6378.1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput: 0.00\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e as immediate Launch is required\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput 47835.75,15,6000,5000 returns -1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=numerology_darryl.cpp.txt\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eThe Judges' E solution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGeometry Hint: Draw a circle and a line from the center of approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46938,"title":"Numerical computation of the optimal shooting angle of a catapult","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 879.667px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.5px 439.833px; transform-origin: 406.5px 439.833px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 32.1667px; text-align: left; transform-origin: 383.5px 32.1667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a capapult that fires a projects into the air with an initial velocity\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"109\" height=\"21\" style=\"width: 109px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and an initial velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, find the optimal shooting angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"18.5\" style=\"width: 16px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eof the catapult that minimizes the distance between the target and the trajectory of the fired projectile. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 21.25px; text-align: left; transform-origin: 383.5px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etip 1:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e Consider the states \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the x- and y-position of the projectile, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45\" height=\"22\" style=\"width: 45px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45.5\" height=\"22\" style=\"width: 45.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e      \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; 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height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"47.5\" height=\"17.5\" style=\"width: 47.5px; height: 17.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eis the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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Plotting \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e vs. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e will result in the x-y trajectory of the projectile, as shown in the figure below.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etip 2: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eUse the following update law, to incrementally update the shooting angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"20\" style=\"width: 15px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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width=\"44\" height=\"20\" style=\"width: 44px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e the smallest Euclidean distance between the trajectory of the projectile and the target \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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\" width=\"298\" height=\"20\" style=\"width: 298px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eis a difference angle, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"55.5\" height=\"17.5\" style=\"width: 55.5px; height: 17.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ean update parameter.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample of algorithm's numerical result:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.5px 40px; transform-origin: 403.5px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003etheta = catapult(25,3,25)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003etheta = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.8431\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 264.333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 383.5px 132.167px; text-align: left; transform-origin: 383.5px 132.167px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"570\" height=\"259\" style=\"vertical-align: baseline;width: 570px;height: 259px\" 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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function theta = catapult(xd,yd,v0) \r\n  \r\n    global g nu;\r\n    \r\n    g   = -9.81;  % grav. acceleration\r\n    nu  = 0.5;    % air friction coeff.\r\n    k   = 0;      % solver increments\r\n    dt  = 1e-2;   % timesteps\r\n    T   = 10;     % simulation time\r\n    TOL = 1e-2;   % absolute tolerance\r\n    \r\n    [~,y] = ode45(@ODECatapult,0:dt:T,[v0,0,0]); \r\n    \r\n    % solver for optimal angle\r\n    while (e \u003e= TOL) \u0026\u0026 (k \u003e 150)        \r\n        \r\n        %theta = theta + beta;\r\n        \r\n        k = k+1;    % add increment\r\n    end\r\n  \r\n    function dx = ODECatapult(t,x)\r\n        global g nu;\r\n        %% fill in ordinary differential equation %%\r\n    end\r\n    \r\n    function e = EuclideanDistance(y,xd,yd)\r\n        %% fill in computation of smallest euclidean distance %%\r\n    end\r\n    \r\n    function beta = UpdateLaw(y,e,lambda)\r\n        %% fill in update law to update the shooting angle %%\r\n    end\r\nend","test_suite":"xd = 8;\r\nyd = 2;\r\nv0 = 35;\r\ny_correct = 1.446;\r\n\r\nassert(isequal(round(catapult(xd,yd,v0),3),y_correct))\r\n\r\n%%\r\nxd = 15;\r\nyd = 5;\r\nv0 = 35;\r\ny_correct = 1.33;\r\n\r\nassert(isequal(round(catapult(xd,yd,v0),2),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":636373,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-19T12:41:43.000Z","updated_at":"2025-01-02T11:31:42.000Z","published_at":"2020-10-19T13:39:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a capapult that fires a projects into the air with an initial velocity\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_0 \\\\in \\\\mathbb{R}_{\\\\ge 0}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$z_d = [x_d, y_d] \\\\in \\\\mathbb{R}^2$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and an initial velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_0 \\\\in \\\\mathbb{R}_{\\\\ge 0}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, find the optimal shooting angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eof the catapult that minimizes the distance between the target and the trajectory of the fired projectile. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etip 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Consider the states \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the x- and y-position of the projectile, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x_1} = x_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,     \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_2 = x_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e      \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_3 = -\\\\nu x_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_4 = -g - \\\\nu x_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg = 9.81\\\\; (\\\\text{m/s}^2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu = 0.5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eis the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t = 0) = (0,0,v_0 \\\\cos(\\\\theta_k), v_0 \\\\sin(\\\\theta_k))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Plotting \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e vs. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will result in the x-y trajectory of the projectile, as shown in the figure below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etip 2: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eUse the following update law, to incrementally update the shooting angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_{k+1} = \\\\theta_k + \\\\lambda \\\\, \\\\text{sign}(\\\\theta_{e,k})\\\\,e_k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee_k \\\\in \\\\mathbb{R}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the smallest Euclidean distance between the trajectory of the projectile and the target \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez_d\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_{e,k} = \\\\text{atan2}(d_y,d_x) - \\\\text{atan2}(v_0\\\\sin(\\\\theta_k),v_0\\\\cos(\\\\theta_k))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eis a difference angle, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\lambda = 0.01$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003ean update parameter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample of algorithm's numerical result:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[theta = catapult(25,3,25)\\ntheta = \\n    0.8431\\n    ]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"259\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"570\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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this tower of blocks going to fall?\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium02.jpg\u003e\u003e\r\n\r\n*Description*\r\n\r\nGiven a stacking configuration for a series of square blocks, your function should return _true_ if they are at equilibrium and _false_ otherwise. \r\n\r\nThe block configuration for N blocks is provided as a input vector *x* with N elements listing the _x-coordinates_ of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The _y-coordinates_ of each block are determined implicitly by the order of the blocks, which are dropped \"tetris-style\" until they hit the floor or another block. \r\n\r\nAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\r\n \r\nIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\r\n\r\n*Examples*:\r\n\r\nExample (1) \r\n \r\n x = [0 0.4];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01a.jpg\u003e\u003e\r\n\r\nThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return _true_.\r\n\r\nExample (2) \r\n\r\n x = [0 0.6];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01b.jpg\u003e\u003e\r\n\r\nThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return _false_.\r\n\r\nExample (3) \r\n\r\n x = [0 1.5 0.6];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01c.jpg\u003e\u003e\r\n\r\nThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return _true_.\r\n\r\nExample (4) \r\n\r\n x = [0 .9 -.9 zeros(1,5)];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01d.jpg\u003e\u003e\r\n\r\nThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\r\n\r\nExample (5) \r\n\r\nx = cumsum(fliplr(1./(1:8))/2);\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01e.jpg\u003e\u003e\r\n\r\nThis configuration is stable (see the \u003chttp://en.wikipedia.org/wiki/Block-stacking_problem classic optimal stacking solution\u003e) so your function should return _true_.\r\n\r\n*Display*\r\n\r\nIf you wish, you may display any given block configuration *x* using the code below:\r\n\r\n clf;\r\n y=[];\r\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u003c1)+1]); end\r\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\r\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\r\n 'horizontalalignment','center');\r\n\r\nVisit \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas Block canvas\u003e for a related Cody problem.","description_html":"\u003cp\u003eIs this tower of blocks going to fall?\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium02.jpg\"\u003e\u003cp\u003e\u003cb\u003eDescription\u003c/b\u003e\u003c/p\u003e\u003cp\u003eGiven a stacking configuration for a series of square blocks, your function should return \u003ci\u003etrue\u003c/i\u003e if they are at equilibrium and \u003ci\u003efalse\u003c/i\u003e otherwise.\u003c/p\u003e\u003cp\u003eThe block configuration for N blocks is provided as a input vector \u003cb\u003ex\u003c/b\u003e with N elements listing the \u003ci\u003ex-coordinates\u003c/i\u003e of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The \u003ci\u003ey-coordinates\u003c/i\u003e of each block are determined implicitly by the order of the blocks, which are dropped \"tetris-style\" until they hit the floor or another block.\u003c/p\u003e\u003cp\u003eAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\u003c/p\u003e\u003cp\u003eIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e:\u003c/p\u003e\u003cp\u003eExample (1)\u003c/p\u003e\u003cpre\u003e x = [0 0.4];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01a.jpg\"\u003e\u003cp\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (2)\u003c/p\u003e\u003cpre\u003e x = [0 0.6];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01b.jpg\"\u003e\u003cp\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return \u003ci\u003efalse\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (3)\u003c/p\u003e\u003cpre\u003e x = [0 1.5 0.6];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01c.jpg\"\u003e\u003cp\u003eThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (4)\u003c/p\u003e\u003cpre\u003e x = [0 .9 -.9 zeros(1,5)];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01d.jpg\"\u003e\u003cp\u003eThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\u003c/p\u003e\u003cp\u003eExample (5)\u003c/p\u003e\u003cp\u003ex = cumsum(fliplr(1./(1:8))/2);\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01e.jpg\"\u003e\u003cp\u003eThis configuration is stable (see the \u003ca href = \"http://en.wikipedia.org/wiki/Block-stacking_problem\"\u003eclassic optimal stacking solution\u003c/a\u003e) so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eDisplay\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIf you wish, you may display any given block configuration \u003cb\u003ex\u003c/b\u003e using the code below:\u003c/p\u003e\u003cpre\u003e clf;\r\n y=[];\r\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u0026lt;1)+1]); end\r\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\r\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\r\n 'horizontalalignment','center');\u003c/pre\u003e\u003cp\u003eVisit \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas\"\u003eBlock canvas\u003c/a\u003e for a related Cody problem.\u003c/p\u003e","function_template":"function y = equilibrium(x)\r\n  y = true;\r\nend","test_suite":"%%\r\nx = [0 0.6];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [0 0.4];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [0 1.5 0.6];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = cumsum(fliplr(1./(1:16))/2);\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [1.5 2.5 1 0.25 5.5 3.5 -1.5 -0.25 -4 -1.75 6.25 -1.25 0.5 1 -0.5 4.75 -1.25 -1.5 0.5 1.5];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [-1.5 -0.25 -0.25 -1.75 1.75 -2 -5.25 2.25 0.75 -0 0.25 0 -1 -1.5 4.75 -1 4.75 -2.5 3.25 -1];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [1.25 -1.75 -0.75 2.75 -1.5 3.75 1.75 1.5 -0.25 -4 -0.5 -2 1.75 -3.5 -2 -3.75 0 -1.75 1.75 3.25 -1.5 -0.5 1.25 -2 1.5 3 -0.25 1.75 -0.5 2.75 0.5 -4.25 1.5 5.5 3 4.25 2.75 -0.75 0.5 3 3.5 3.25 1.75 1.5 3.25 2.5 5.5 -2 -3.75 -1 5 0.25 -3.75 5.5 1.75 2 1.75 0.5 -3.75 1.5 -0 2 0.5 0 -0.25 0.25 -9 1.75 -3.75 1.25 -3.75 -0 1.75 3.5 -3.75 3.75 -5.75 1.25 3.5 1.5 -2 2 -2.5 -1.5 3 1.75 -1.5 3.25 1.75 -1.5 1.25 -2.5 1.25 -4.25 3.25 -2.5 1.75 1.75 7.25 3.5];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [-0.5 1.5 3.5 0.5 -5.75 2 -1.5 0.25 -0.25 -3 -0.25 3 -3.5 -4.5 1.75 -0.25 0.75 3 0.25 -2.5 2.25 -0.25 1.75 -1.5 -5 -0 -0.5 -2 -0 4.75 -0 2 3.5 1.5 -2.25 3.5 -0.5 4.5 2.5 0.5 1.75 3.5 -0 -2.25 -0.25 -4.75 -2.5 -0.75 -6 2.75 -5 2.25 1 -2.25 -0.75 -0.25 -3.5 0.75 -0 -0.5 -0.5 -1.75 -2 -0.25 -0.25 5 -0.25 -0.75 -0.25 -5 -2 -0.25 -5.5 -5 -0.5 -2 1 -0.75 2 3.25 4.5 2.25 1.25 -0.25 -0.5 -0.25 -2.5 -5 2.25 -2 7.5 6.5 2.25 -0.25 -0.5 7.25 -2.5 1 -2.5 -4.75];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [0.1 0.1 -4.6 -0.4 -1.5 1.6 3 2.7 2.3 -2.7 0.1 -1.7 0 4.4 3.8 -0.4 -2 -0.6 3.3 2.5 -3 -1.7 3.1 2.7 2.7 3.1 -0.4 1.1 -0.2 -0.1 -0.3 2.7];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [0.1 0.1 -4.6 -0.4 -1.5 1.6 3 2.7 2.3 -2.7 0.1 -1.7 0 4.4 3.8 -0.4 -2 -0.6 3.3 2.5 -3 -1.7 3.1 2.7 2.7 3.1 -0.4 1.1 -0.2 -0.1 -0.3 2.7 -1.8 2.3];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [-3.5 2.6 -0.7 -1.1 -2.6 -2 0.8 -0.7 2.7 0.4 -5 3.7 -1.2 -1.3 2.8 0.8 -1.5 -1.8 0 -0 4.8 -1.4 -1.2 -1.5 1 0.2 2.6 1.7 1.6 -1.3 2.1 -1.5 -1.4 2 0.1 -0.1 -0.1 4.6 -3 -0.3 0.2 -1.9 -0 0.1 0 2.1 -1.7 -3.1 -0 0.2 -0.1 -0.5 4.7 -1.8 -0.1 -2.2];\r\nassert(isequal(equilibrium(x),false));\r\n\r\n%%\r\nx = [-2.5 2.6 -0.7 -1.1 -2.6 -2 0.8 -0.7 2.7 0.4 -5 3.7 -1.2 -1.3 2.8 0.8 -1.5 -1.8 0 -0 4.8 -1.4 -1.2 -1.5 1 0.2 2.6 1.7 1.6 -1.3 2.1 -1.5 -1.4 2 0.1 -0.1 -0.1 4.6 -3 -0.3 0.2 -1.9 -0 0.1 0 2.1 -1.7 -3.1 -0 0.2 -0.1 -0.5 4.7 -1.8 -0.1 -2.2];\r\nassert(isequal(equilibrium(x),true));\r\n\r\n%%\r\nx =[0 .9 -.9 zeros(1,8)];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx =[0 .9 -.9 zeros(1,6)];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = repmat([0 .7 -.7 0],1,2);\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = repmat([0 .6 -.6 0],1,2);\r\nassert(isequal(equilibrium(x),true))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":43,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-10-02T08:49:59.000Z","updated_at":"2013-10-08T00:22:34.000Z","published_at":"2013-10-02T09:44:07.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId3\",\"target\":\"/media/image3.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId4\",\"target\":\"/media/image4.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId5\",\"target\":\"/media/image5.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId6\",\"target\":\"/media/image6.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIs this tower of blocks going to fall?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDescription\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a stacking configuration for a series of square blocks, your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if they are at equilibrium and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efalse\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e otherwise.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe block configuration for N blocks is provided as a input vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with N elements listing the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex-coordinates\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey-coordinates\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of each block are determined implicitly by the order of the blocks, which are dropped \\\"tetris-style\\\" until they hit the floor or another block.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 0.4];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 0.6];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efalse\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 1.5 0.6];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 .9 -.9 zeros(1,5)];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = cumsum(fliplr(1./(1:8))/2);\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis configuration is stable (see the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Block-stacking_problem\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eclassic optimal stacking solution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e) so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDisplay\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf you wish, you may display any given block configuration\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e using the code below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ clf;\\n y=[];\\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u003c1)+1]); end\\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\\n 'horizontalalignment','center');]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eVisit\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eBlock canvas\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for a related Cody problem.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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Graph: Wichmann Rulers","description":"This Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Graceful Graph Contest\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\r\n\r\nAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003chttp://oeis.org/A193802 Optimal Wichmann Ruler\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\r\n\r\nThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\r\n\r\nFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\r\n\r\n*Input:* P  (Number of Points on the ruler)\r\n\r\n*Output:* S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\r\n\r\n*Notes:*\r\n\r\n  1) A W(r,s) does not guarantee all deltas can be generated\r\n  2) For any P there are multiple W(r,s) solutions \r\n  3) P=5 solution is 9, readily solved by brute force\r\n  4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n  5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun ","description_html":"\u003cp\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003eGraceful Graph Contest\u003c/a\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\u003c/p\u003e\u003cp\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003ca href = \"http://oeis.org/A193802\"\u003eOptimal Wichmann Ruler\u003c/a\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/p\u003e\u003cp\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\u003c/p\u003e\u003cp\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e P  (Number of Points on the ruler)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/p\u003e\u003cp\u003e\u003cb\u003eNotes:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e1) A W(r,s) does not guarantee all deltas can be generated\r\n2) For any P there are multiple W(r,s) solutions \r\n3) P=5 solution is 9, readily solved by brute force\r\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun \r\n\u003c/pre\u003e","function_template":"function s=Graceful_Wichmann(n)\r\n  s=0;\r\nend","test_suite":"%%\r\ntic\r\nn=17;\r\nexp=101;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=19;\r\nexp=123;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=23;\r\nexp=183;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=29;\r\nexp=289;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=31;\r\nexp=327;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=37;\r\nexp=465;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=41;\r\nexp=573;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=43;\r\nexp=627;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=47;\r\nexp=751;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=53;\r\nexp=953;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=59;\r\nexp=1179;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=61;\r\nexp=1257;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=67;\r\nexp=1515;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=71;\r\nexp=1703;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=73;\r\nexp=1797;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=79;\r\nexp=2103;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=83;\r\nexp=2323;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=89;\r\nexp=2669;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=97;\r\nexp=3165;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-23T01:30:25.000Z","updated_at":"2013-09-23T13:04:40.000Z","published_at":"2013-09-23T04:00:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u0026gt;13. This Challenge is related to the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u0026gt;13.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points. An\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A193802\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOptimal Wichmann Ruler\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u0026gt;=0 and integer).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P (Number of Points on the ruler)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNotes:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[1) A W(r,s) does not guarantee all deltas can be generated\\n2) For any P there are multiple W(r,s) solutions \\n3) P=5 solution is 9, readily solved by brute force\\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":49718,"title":"Malus’ Law (theta) between polarizations","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 20.736px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.5px 10.368px; transform-origin: 406.5px 10.368px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.496px 10.368px; text-align: left; transform-origin: 383.502px 10.368px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the angle (\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eθ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e theta) ,I1 and I2 will be given \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = t(x,y)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = 10;y=5\r\ny_correct = 60;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=5\r\ny_correct = 75.5225;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=10\r\ny_correct = 60;\r\nassert(isequal(t(x,y),y_correct))\r\n%%\r\nx = 20;y=4\r\ny_correct = 78.4630;\r\nassert(isequal(t(x,y),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":610936,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-31T03:26:31.000Z","updated_at":"2026-04-02T12:26:44.000Z","published_at":"2020-12-31T03:44:26.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the angle (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e theta) ,I1 and I2 will be given \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":61185,"title":"Compute wheel slip ratio during braking.","description":"During braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as wheel slip ratio, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\r\nGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(33, 33, 33); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-inline-start: 2px; padding-left: 2px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 383px 31.5px; text-align: left; transform-origin: 383px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eDuring braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ewheel slip ratio\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 383px 21px; text-align: left; transform-origin: 383px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function s = wheelSlipRatio(v_vehicle, v_wheel)\r\ns = 0;\r\nend\r\n","test_suite":"%%\r\nv_vehicle = 20; v_wheel = 18;\r\ns_correct = 0.1;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n%%\r\nv_vehicle = 25; v_wheel = 0;\r\ns_correct = 1;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n%%\r\nv_vehicle = 0; v_wheel = 0;\r\ns_correct = 0;\r\nassert(abs(wheelSlipRatio(v_vehicle,v_wheel) - s_correct) \u003c 1e-6)\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":2305225,"edited_by":2305225,"edited_at":"2026-02-02T06:36:51.000Z","deleted_by":null,"deleted_at":null,"solvers_count":32,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2026-02-02T06:36:05.000Z","updated_at":"2026-04-04T03:41:48.000Z","published_at":"2026-02-02T06:36:51.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDuring braking, a difference develops between the vehicle’s forward speed and the rotational speed of its wheels. This difference is captured by a quantity known as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ewheel slip ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, which plays a critical role in traction control, ABS algorithms, and vehicle stability systems.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the vehicle longitudinal speed and wheel circumferential speed, determine the corresponding slip ratio. Your implementation should handle normal driving conditions, heavy braking scenarios, and boundary cases reliably.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":59711,"title":" Calculating distance of lightning based on time delay","description":"If we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 42px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 21px; transform-origin: 407px 21px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIf we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = your_fcn_name(t)\r\n  %Easier than it looks\r\nend","test_suite":"%%\r\nt = 1;\r\ny_correct = 343;\r\nassert(isequal(your_fcn_name(t),y_correct))\r\n%%\r\nt = 57;\r\ny_correct = 19551;\r\nassert(isequal(your_fcn_name(t),y_correct))\r\n%%\r\nt = randi(1000);\r\ny_correct = 343*t;\r\nassert(isequal(your_fcn_name(t),y_correct))","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":4207616,"edited_by":4207616,"edited_at":"2024-03-23T10:18:40.000Z","deleted_by":null,"deleted_at":null,"solvers_count":13,"test_suite_updated_at":"2024-03-23T10:18:40.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-03-23T10:15:53.000Z","updated_at":"2025-08-26T12:10:35.000Z","published_at":"2024-03-23T10:18:40.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf we know the time delay between when we see lightning and hear thunder then we can calculate approximate distance(in meters) of how far the incident happened. Write a code to calculate that distance if the time delay t is known. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":42689,"title":"Distance a ball travels after throwing vertically","description":"Calculate the total distance *'d'* (in meters) a ball would travel after *'s'* seconds and starting velocity of *'v'* (in m/s). \"Please note that its not relative distance but total distance travelled\"\r\n\r\ngravity=10 m/s^2\r\n\r\nExample: initial speed = +20, how long does the ball travel after 2.5 sec\r\nd=21.25","description_html":"\u003cp\u003eCalculate the total distance \u003cb\u003e'd'\u003c/b\u003e (in meters) a ball would travel after \u003cb\u003e's'\u003c/b\u003e seconds and starting velocity of \u003cb\u003e'v'\u003c/b\u003e (in m/s). \"Please note that its not relative distance but total distance travelled\"\u003c/p\u003e\u003cp\u003egravity=10 m/s^2\u003c/p\u003e\u003cp\u003eExample: initial speed = +20, how long does the ball travel after 2.5 sec\r\nd=21.25\u003c/p\u003e","function_template":"function d = distance(s,v)\r\nif v-10*s\u003e0\r\nd=((v-s*10)+v)/2*s\r\nelseif 2*v-10*s\u003c0\r\nd=v*v/10\r\nelse\r\nd=v/10*v/2+(s-v/10)^2*10/2\r\nend\r\nend","test_suite":"%%\r\nv = 20;\r\ns=2;\r\nd = 20;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv = 20;\r\ns=2.5;\r\nd = 21.25;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv = 20;\r\ns=12.5;\r\nd=40;\r\nassert(isequal(distance(s,v),d));\r\n%%\r\n\r\nv=50;\r\ns=10;\r\nd=250;\r\n\r\nv=5;\r\ns=5;\r\nd=2.5;\r\nassert(isequal(distance(s,v),d));","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":9199,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":57,"test_suite_updated_at":"2015-11-25T15:32:59.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2015-11-25T11:24:15.000Z","updated_at":"2026-02-04T17:51:38.000Z","published_at":"2015-11-25T15:29:05.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the total distance\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e'd'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (in meters) a ball would travel after\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e's'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e seconds and starting velocity of\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e'v'\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (in m/s). \\\"Please note that its not relative distance but total distance travelled\\\"\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003egravity=10 m/s^2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample: initial speed = +20, how long does the ball travel after 2.5 sec d=21.25\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":48685,"title":"Laws of motion 3","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 21px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 10.5px; transform-origin: 407px 10.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind the final velocity of an object with initial velocity 'u', acceleration 'a' and distance travelled 's'. Round to nearest number.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function v = your_fcn_name(u,a,s)\r\n  y = x;\r\nend","test_suite":"%%\r\nu=0;\r\na=1;\r\ns=1;\r\ny_correct = 1;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=0;\r\na=1;\r\ns=10;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=20;\r\na=2;\r\ns=10;\r\ny_correct = 21;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n\r\n%%\r\nu=0;\r\na=1;\r\ns=7;\r\ny_correct = 4;\r\nassert(isequal(your_fcn_name(u,a,s),y_correct))\r\n","published":true,"deleted":false,"likes_count":24,"comments_count":5,"created_by":644918,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3285,"test_suite_updated_at":"2020-12-21T16:57:51.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-21T16:53:43.000Z","updated_at":"2026-04-04T03:59:26.000Z","published_at":"2020-12-21T16:53:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind the final velocity of an object with initial velocity 'u', acceleration 'a' and distance travelled 's'. Round to nearest number.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":441,"title":"Drying sweater?","description":"* A sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes. \r\n* Veronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer. \r\n* How many minutes she should wait so that the sweater weighs w times its dry weight? \r\n* Please try a general solution, the test suite may expand later.","description_html":"\u003cul\u003e\u003cli\u003eA sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes.\u003c/li\u003e\u003cli\u003eVeronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer.\u003c/li\u003e\u003cli\u003eHow many minutes she should wait so that the sweater weighs w times its dry weight?\u003c/li\u003e\u003cli\u003ePlease try a general solution, the test suite may expand later.\u003c/li\u003e\u003c/ul\u003e","function_template":"function m = drying(x,w)\r\n   m=x/w;\r\nend","test_suite":"%%\r\nx=2; w=1.5; m=drying(x,w);\r\nm_correct = 69;\r\nassert(round(m)==m_correct)\r\n%%\r\nx=3; w=2; m=drying(x,w);\r\nm_correct = 69;\r\nassert(round(m)==m_correct)\r\n%%\r\nx=5; w=2; m=drying(x,w);\r\nm_correct = 139;\r\nassert(round(m)==m_correct)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":166,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":33,"test_suite_updated_at":"2012-03-06T17:52:02.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-02T17:09:51.000Z","updated_at":"2025-05-13T14:59:54.000Z","published_at":"2012-03-13T16:01:43.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA sweater is revolving in a slow low power dryer and losing moisture at any moment at the constant rate 100% of its current moisture content in 100 minutes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eVeronica knew the sweater was x times its dry weight when she put her wet sweater from the washer to the dryer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHow many minutes she should wait so that the sweater weighs w times its dry weight?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePlease try a general solution, the test suite may expand later.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":480,"title":"Aufbau principle ","description":"Given the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\r\nsum(x)==sumofx\r\nlength of x is the shortest possible\r\nx has positive integers only\r\nif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\r\nif x(k+1)\u003e0 then x(k) must be maximum possible.\r\nFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info: Aufbau Principle, Electron Shell.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 216.167px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 108.083px; transform-origin: 407px 108.083px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363.5px 8px; transform-origin: 363.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 51.0833px; transform-origin: 391px 51.0833px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 51px 8px; transform-origin: 51px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esum(x)==sumofx\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 107px 8px; transform-origin: 107px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003elength of x is the shortest possible\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003ex has positive integers only\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 246px 8px; transform-origin: 246px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 150.5px 8px; transform-origin: 150.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eif x(k+1)\u0026gt;0 then x(k) must be maximum possible.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eAufbau Principle\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eElectron Shell\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function electrons = Aufbau(sumofx)\r\nif sumofx==3; \r\n   x=[2 1]; \r\n   electrons='1s2,2s1'; \r\nend\r\n","test_suite":"%%\r\nsumofx = 3; % Lithium\r\nelectrons = '1s2,2s1';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 6; % Carbon\r\nelectrons = '1s2,2s2,2p2';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 10; % Neon\r\nelectrons = '1s2,2s2,2p6';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 17; % Chlorine\r\nelectrons = '1s2,2s2,2p6,3s2,3p5';\r\nassert(isequal(electrons,Aufbau(sumofx)))\r\n%%\r\nsumofx = 20; % Chlorine\r\nelectrons = '1s2,2s2,2p6,3s2,3p6,4s2';\r\nassert(isequal(electrons,Aufbau(sumofx)))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":166,"edited_by":223089,"edited_at":"2022-10-29T14:54:34.000Z","deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2022-10-29T14:54:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-12T03:36:58.000Z","updated_at":"2026-01-02T13:13:55.000Z","published_at":"2012-03-12T15:44:42.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the order e=[1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p], find a vector x with these conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esum(x)==sumofx\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003elength of x is the shortest possible\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex has positive integers only\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif e(k) contains s,p,d,f,g, then x(k) must be less than 3,7,11,15,19, respectively\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eif x(k+1)\u0026gt;0 then x(k) must be maximum possible.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if sumofx=3 then x=[2 1]. Return x embedded in e in the following style: electrons='1s2,2s1'. Please see more info:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eAufbau Principle\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eElectron Shell\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44936,"title":"Float like a cannonball","description":"Given gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative).\r\nHint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!","description_html":"\u003cp\u003eGiven gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative).\r\nHint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!\u003c/p\u003e","function_template":"function s = CannonBall(u,theta)\r\n%Stones taught me to fly...\r\n  s = u*theta;\r\nend","test_suite":"%%\r\nu= 100;\r\ntheta=85;\r\ny_correct = 177;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 31.42;\r\ntheta=45;\r\ny_correct = 101;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 31.42;\r\ntheta=-41;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=30;\r\ny_correct = 883;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= -100;\r\ntheta=30;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n\r\n%%\r\nu= -100;\r\ntheta=210;\r\ny_correct = 883;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n\r\n%%\r\nu= 100;\r\ntheta=210;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 0;\r\ntheta=40;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=90;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nu= 100;\r\ntheta=0;\r\ny_correct = 0;\r\nassert(isequal(round(CannonBall(u,theta)),y_correct))\r\n\r\n%%\r\nfiletext = fileread('CannonBall.m');\r\nassert(isempty(strfind(filetext, 'regexp'))); assert(isempty(strfind(filetext, 'eval'))) \r\n\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":170350,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":26,"test_suite_updated_at":"2019-08-02T09:56:25.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2019-08-02T09:45:02.000Z","updated_at":"2026-01-02T13:04:50.000Z","published_at":"2019-08-02T09:45:02.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven gravity on earth (g=9.81 [m/s/s]) find the distance s [m] by a cannonball propelled at a speed of u [m/s] from the origin at an angle theta [deg] measured from the horizontal. Assume no air resistance or bouncing. The altitude of the release of the cannon ball is 0 m and the travel should have the appropriate sign (i.e. behind the release point would be negative). Hint: Consider logical reasoning when the orientation of the firing vector would be into the ground!!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":47435,"title":"Wavy gravity","description":"In a parallel universe the gravity works very strangely.\r\nIndeed, gravity is equal to: g = sin( (pi/2)*t/60 ) [m/s^2], where t is the time expressed in seconds, and it acts along the direction normal to the ground.\r\nAt time t = 0 seconds, a ball is at height y = 1 meter and has zero velocity. What is the position y (meters) of the ball as a function of time (seconds)?","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 123px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 61.5px; transform-origin: 407px 61.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.5px 8px; transform-origin: 168.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn a parallel universe the gravity works very strangely.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIndeed, gravity is equal to: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eg\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105.5px 8px; transform-origin: 105.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = sin( (pi/2)*t/60 ) [m/s^2], where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003et\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.5px 8px; transform-origin: 174.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the time expressed in seconds, and it acts along the direction normal to the ground.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23.5px 8px; transform-origin: 23.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAt time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003et\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 0 seconds, a ball is at height \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168px 8px; transform-origin: 168px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = 1 meter and has zero velocity. What is the position \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ey\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79px 8px; transform-origin: 79px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (meters) of the ball as a function of time (seconds)?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function yf = parUnig(tf)\r\ny = [];\r\nend","test_suite":"%%\r\ntf = 30; % seconds\r\ny_correct = 115; % meters\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n\r\n%%\r\ntf = 60;\r\ny_correct = 834;\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n\r\n%%\r\ntf = 120;\r\ny_correct = 4585;\r\nassert(isequal(round(parUnig(tf)), y_correct))\r\n","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":280347,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":"2022-01-01T10:27:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-11-10T11:04:29.000Z","updated_at":"2022-01-01T10:27:26.000Z","published_at":"2020-11-10T11:06:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn a parallel universe the gravity works very strangely.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIndeed, gravity is equal to: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = sin( (pi/2)*t/60 ) [m/s^2], where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the time expressed in seconds, and it acts along the direction normal to the ground.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAt time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 0 seconds, a ball is at height \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = 1 meter and has zero velocity. What is the position \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (meters) of the ball as a function of time (seconds)?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":46968,"title":"Electric Potential Energy","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 132px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 66px; transform-origin: 407px 66px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCalculate the magnitude of Electric potential energy between two given charges [q1, q2] with spatial coordinates in the form [x1 y1; x2 y2]. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTake the value of absolute dielectric permittivity 1e-9/(36pi).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAll units are SI units.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eUse of if-else is prohibited to prevent hard-coded solutions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function F = ESF(q, c)\r\nq,c;\r\nend","test_suite":"%%\r\nfiletext = fileread('ESF.m');\r\nassert(isempty(strfind(filetext, 'if ')))\r\n\r\n%%\r\nq = [1e-5 1e-5];\r\nc = [1 1; 4 5];\r\n\r\nassert(abs(ESF(q,c)-0.18)\u003c1e-6);\r\n\r\n%%\r\nq = [1e-7 1e-6];\r\nc = [0 0; -6 -8];\r\n\r\nassert(abs(ESF(q,c)-9e-5)\u003c1e-6);\r\n\r\n%%\r\nq = [1e-2 -1e-3];\r\nc = [0.5 0.75; -2.5 -2.25]/sqrt(2);\r\n\r\nassert(abs(ESF(q,c)+3e4)\u003c1e-6);\r\n\r\n%%\r\nq = [-1 -1];\r\nc = [-3 -4;-3 -1];\r\n\r\nassert(abs(ESF(q,c)-3e9)\u003c1e-6);\r\n\r\n%%\r\nq = [1e5 1e-15];\r\nc = [-0 +0; -200 990];\r\n\r\nassert(abs(ESF(q,c)-8.911e-4)\u003c1e-6);\r\n\r\n%%\r\nq = [-1e-7 1e-13];\r\nc = [-27 100; 92 -220];\r\n\r\nassert(abs(ESF(q,c)+2.6361e-13)\u003c1e-6);\r\n\r\n%%\r\nq = [1e30 1e-30];\r\nc = [-exp(1) pi; -exp(1) pi+1];\r\n\r\nassert(abs(ESF(q,c)-9e9)\u003c1e-6);","published":true,"deleted":false,"likes_count":2,"comments_count":9,"created_by":223089,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":14,"test_suite_updated_at":"2020-12-27T17:53:49.000Z","rescore_all_solutions":true,"group_id":1,"created_at":"2020-10-20T11:19:48.000Z","updated_at":"2025-04-02T17:27:01.000Z","published_at":"2020-10-20T11:19:48.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCalculate the magnitude of Electric potential energy between two given charges [q1, q2] with spatial coordinates in the form [x1 y1; x2 y2]. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTake the value of absolute dielectric permittivity 1e-9/(36pi).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll units are SI units.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUse of if-else is prohibited to prevent hard-coded solutions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":54710,"title":"Compute the period of a pendulum started from a finite initial angle","description":"Cody Problem 49830 asks for the period  of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle  that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \r\nWrite a function that takes the initial angle and returns , where  is the length of the pendulum and  is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 94.1px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 47.05px; transform-origin: 407px 47.05px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 43px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.5px; text-align: left; transform-origin: 384px 21.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/49830\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eCody Problem 49830\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.45px 8px; transform-origin: 61.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e asks for the period \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eT\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 240.792px 8px; transform-origin: 240.792px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"theta0\" style=\"width: 15px; height: 20px;\" width=\"15\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 306.883px 8px; transform-origin: 306.883px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.1px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.05px; text-align: left; transform-origin: 384px 21.05px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 168.683px 8px; transform-origin: 168.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function that takes the initial angle and returns \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"T\\sqrt{g/L}\" style=\"width: 52.5px; height: 20.5px;\" width=\"52.5\" height=\"20.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8917px 8px; transform-origin: 24.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 107.358px 8px; transform-origin: 107.358px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the length of the pendulum and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eg\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"2pi\" style=\"width: 19.5px; height: 18px;\" width=\"19.5\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pendulumPeriod(theta0)\r\n  T = theta0-theta0^3/3!+theta0^5/5!+higher order terms;\r\nend","test_suite":"%%\r\nth = pi/7;\r\nT_correct = 6.363207946270837;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/5;\r\nT_correct = 6.44181661515865;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% Problem 1 on p. 194 of Davis (1962)\r\nth = pi/4;\r\nT_correct = 6.534345229832591;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/3;\r\nT_correct = 6.743001419250384;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = pi/2;\r\nT_correct = 7.416298709205487;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nth = 36*pi/37;\r\nT_correct = 18.190113206504414;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 72*pi/73;\r\nT_correct = 20.902949604823448;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%% \r\nth = 0;\r\nT_correct = 2*pi;\r\nassert(abs(pendulumPeriod(th)-T_correct)\u003c1e-12)\r\n\r\n%%\r\nfiletext = fileread('pendulumPeriod.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'switch'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":46909,"edited_by":46909,"edited_at":"2022-06-06T01:13:14.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-06-06T00:37:45.000Z","updated_at":"2026-01-09T20:11:24.000Z","published_at":"2022-06-06T01:13:14.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/49830\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody Problem 49830\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e asks for the period \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e of a pendulum swinging through a small angle. Here the pendulum started at rest from an angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"theta0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that is not necessarily small. The other assumptions are similar (no friction or drag, massless rod). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function that takes the initial angle and returns \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"T\\\\sqrt{g/L}\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eT\\\\sqrt{g/L}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"L\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the length of the pendulum and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"g\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the acceleration of gravity. In the limit as the initial angle approaches zero, the function should produce \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"2pi\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e2\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":433,"title":"jogging?","description":"Imagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you.  You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.","description_html":"\u003cp\u003eImagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you.  You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.\u003c/p\u003e","function_template":"function degrees = jogging(d,r,s)\r\ndegrees=0;\r\nend","test_suite":"%%\r\nd = 4; r = 1; s = 1; degrees_correct = 60;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 5; r = 1; s = 1; degrees_correct = 66;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 6; r = 1; s = 1; degrees_correct = 71;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 7; r = 1; s = 1; degrees_correct = 73;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 8; r = 1; s = 1; degrees_correct = 76;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n%%\r\nd = 7; r = 1; s = 2; degrees_correct = 55;\r\ndegrees = round(jogging(d,r,s));\r\nassert(degrees==degrees_correct)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":3,"created_by":166,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":40,"test_suite_updated_at":"2012-03-02T10:41:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-03-01T21:44:31.000Z","updated_at":"2026-01-31T12:59:31.000Z","published_at":"2012-03-02T16:52:52.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eImagine x-y coordinate system and you are at the origin and your partner is on the x-axis at some small distance (d) away from you. You and your partner started jogging simultaneously, left foot first. You and your partner maintained identical rate of steps per second (r) and length of every step (s). Your track and your partner's track are parallel but at angle alpha to x-axis. Please note that alpha is less than 90 degrees. At the initial moment of starting your partner appeared to be ahead of you due to the angle alpha not being 90 degrees. After you finished jogging a long distance, a photograph of your foot prints at a random location looked as if your partner was neither behind nor ahead of you. Please calculate the largest alpha (degrees) possible under these constraints.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":52654,"title":"Easy Sequences 13: Average Speed of Spaceship","description":"A certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around. \r\nGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops. Please round-off your answer to the nearest integer.\r\nNOTE: Use clasical physics only. Ignore any relativistic effects.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440000534057617px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: normal; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 165px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eA certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"font-weight: bold; \"\u003eGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops.\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003e Please round-off your answer to the nearest integer.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; text-align: left; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; \"\u003e\u003cspan style=\"\"\u003eNOTE: Use clasical physics only. Ignore any relativistic effects.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = mean_velocity(s,v)\r\n  y = x;\r\nend","test_suite":"%%\r\ns = 10000;\r\nv = 10000;\r\nv_correct = 1022;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = 1234567;\r\nv = 1234567;\r\nv_correct = 84539;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = '1234567891011121314151617181920';\r\nv = 123456789;\r\nv_correct = 6427156;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = 1e100;\r\nvs = 1:1000;\r\nv_correct = 72076;\r\nassert(isequal(sum(arrayfun(@(v) mean_velocity(s,v),vs)),v_correct))\r\n%%\r\ns = intmax;\r\nv = double(intmax);\r\nv_correct = 97326319;\r\nassert(isequal(mean_velocity(s,v),v_correct))\r\n%%\r\ns = intmax('int64')/100;\r\nv = double(intmax('int64'))/100;\r\nv_correct = 2326765408587627;\r\nassert(isequal(mean_velocity(s,v),v_correct))","published":true,"deleted":false,"likes_count":2,"comments_count":4,"created_by":255988,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":17,"test_suite_updated_at":"2021-09-05T14:22:29.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2021-09-05T08:20:36.000Z","updated_at":"2025-12-22T16:16:27.000Z","published_at":"2021-09-05T08:20:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA certain alien spaceship is capable of traveling at extremely high velocities and is able to change speed instantaneously. The ship travels from two points 's' km apart, at a speed of 'v' km/hr. After reaching its destination the spaceship immediately heads back to its starting point at the speed of 'v-1' km.hr. After reaching the starting point it again goes back at a speed of 'v-2'. This \\\"back and forth' continues, reducing the ship's speed by 1 km/hr in each turn around.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven an integer initial velocity, find the average speed of the spaceship througout its entire journey, until it finally stops.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Please round-off your answer to the nearest integer.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNOTE: Use clasical physics only. Ignore any relativistic effects.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60326,"title":"Determine the mass of a bat which strikes a ball","description":"Given the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\r\nRound everything to 2 decimals.\r\nvariables: v, f, s, phi","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 102px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 51px; transform-origin: 407px 51px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eRound everything to 2 decimals.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003evariables: v, f, s, phi\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function m = mass(s,f,v,phi)\r\n  E = m*c^2;\r\nend","test_suite":"%%\r\ns = 2.23;\r\nf = 400\r\nv = 115/3.6\r\nphi = 10\r\ng = 9.81\r\ny_correct = 1.74;\r\nassert(isequal(mass(s,f,v,phi),y_correct))\r\n\r\n%%\r\ns = 2.23;\r\nf = 500\r\nv = 115/3.6\r\nphi = 10\r\ng = 9.81\r\ny_correct = 2.17;\r\nassert(isequal(mass(s,f,v,phi),y_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":4201666,"edited_by":4201666,"edited_at":"2024-05-16T16:54:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":"2024-05-16T16:54:48.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2024-05-15T21:31:53.000Z","updated_at":"2024-05-16T16:54:48.000Z","published_at":"2024-05-15T21:31:53.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven the speed, the force of the impact and the length of the trajectory calculate the mass of the bat which the player holds provided the trajectory is a straight line at a 10 degree angle downwards.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRound everything to 2 decimals.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003evariables: v, f, s, phi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58778,"title":"Count collisions in an idealized block system","description":"Two blocks, which have masses  and , slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed . All of the collisions are elastic. \r\nWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \r\n\r\n","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 358.7px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 179.35px; transform-origin: 407px 179.35px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 101.525px 8px; transform-origin: 101.525px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTwo blocks, which have masses \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 19px; height: 20px;\" width=\"19\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 231.058px 8px; transform-origin: 231.058px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 14.5px; height: 20px;\" width=\"14.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.1833px 8px; transform-origin: 99.1833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. All of the collisions are elastic. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 371.725px 8px; transform-origin: 371.725px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 203.7px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 101.85px; text-align: left; transform-origin: 384px 101.85px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline;width: 328px;height: 198px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\" width=\"328\" height=\"198\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = countCollisions(m1,m2,v1)\r\n  y = length(ode45('NewtonII',tspan,y0));\r\n  ","test_suite":"%%\r\nm1 = 1;\r\nm2 = 1;\r\nv1 = 1;\r\ny_correct = 3;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 2;\r\nm2 = 1;\r\nv1 = 1;\r\ny_correct = 5;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 17;\r\nm2 = 4;\r\nv1 = 8;\r\ny_correct = 6;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 15;\r\nm2 = 1.5;\r\nv1 = 0.3;\r\ny_correct = 10;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 3;\r\nm2 = 0.03;\r\nv1 = 0.5;\r\ny_correct = 31;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 4500000;\r\nm2 = 4.5;\r\nv1 = 0.55;\r\ny_correct = 3141;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nm1 = 1.8e11;\r\nm2 = 18;\r\nv1 = 5.2;\r\ny_correct = 314159;\r\ny = countCollisions(m1,m2,v1);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\np2 = regexprep('11.0010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000','\\.','');\r\nfor k = 2:10\r\n    m2 = rand;\r\n    m1 = 4^k*m2;\r\n    v1 = rand;\r\n    y = countCollisions(m1,m2,v1);\r\n    assert(isequal(dec2bin(y),p2(1:k+2)))\r\nend\r\n\r\n%%\r\np7 = regexprep('3.0663651432036134110263402244652226643520650240155443215426431025161154565220002622436103301443233631','\\.','');\r\nfor k = 2:8\r\n    m2 = rand;\r\n    m1 = 49^k*m2;\r\n    v1 = rand;\r\n    y = countCollisions(m1,m2,v1);\r\n    assert(isequal(dec2base(y,7),p7(1:k+1)))\r\nend\r\n\r\n%%\r\nfiletext = fileread('countCollisions.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert') || contains(filetext, 'classdef'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-07-22T01:52:49.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-22T01:47:49.000Z","updated_at":"2023-07-22T01:52:49.000Z","published_at":"2023-07-22T01:47:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwo blocks, which have masses \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, slide along a frictionless, semi-infinite track bounded by a stationary wall. Initially block 2 is not moving, and block 1 moves to the left with speed \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. All of the collisions are elastic. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to count the collisions between the two blocks and block 2 and the wall. For example, if the blocks each have mass of 1 kg and block 1 initially moves to the left at 0.5 m/s, there will be three collisions: the initial collision, a collision between block 2 and the wall, and a final collision between the blocks. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"198\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"328\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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Center of Mass for a Set of Floating Spheres ","description":"Each sphere has a position determined by theta (x,y plane angle) and tau (elevation angle) as well as L, the distance of the center of the sphere from the origin. Each sphere also has a radius, r, and a density of rho. \r\nThese values are defined in a single input matrix: sceneAttributes\r\nThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\r\nAll angles are in degrees, all distances are in meters, and density is in kg/m^3\r\nAssume density and lengths are always positive","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 162px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 81px; transform-origin: 407px 81px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEach sphere has a position determined by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etheta\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (x,y plane angle) and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etau\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (elevation angle) as well as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eL\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the distance of the center of the sphere from the origin. Each sphere also has a radius, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003er\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and a density of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003erho\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThese values are defined in a single input matrix: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003esceneAttributes\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAll angles are in degrees, all distances are in meters, and density is in kg/m^3\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume density and lengths are always positive\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function pos = sphereCOM(sceneAttributes)\r\n  y = x;\r\nend","test_suite":"%%\r\nx = [45, 45, 3, 0.5, 2];\r\ny_correct = [1.5 1.5 2.1213];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\nx = [45, 45, 3, 0.5, 2;\r\n     45, -135, 3, 0.5, 2];\r\ny_correct = [0, 0, 0];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\n%%\r\nx = [45, 45, 3, 0.5, 2;\r\n     135, 45, 3, 0.5, 2;\r\n     225, 45, 3, 0.5, 2;\r\n     315, 45, 3, 0.5, 2];\r\ny_correct = [0, 0, 2.1213];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\nx = [82.0000   70.0000   37.0000    0.3567    5.0000\r\n  -84.0000   69.0000   93.0000    0.5793    2.0000\r\n   11.0000   50.0000   85.0000    0.0155    3.0000\r\n -227.0000   64.0000   18.0000    0.0468    5.0000\r\n -133.0000  -74.0000   10.0000    0.9638    5.0000\r\n -354.0000  -43.0000   86.0000    0.6238    1.0000\r\n   83.0000   32.0000   35.0000    0.9870    5.0000\r\n -293.0000   79.0000    6.0000    0.8137    4.0000\r\n  109.0000   35.0000   62.0000    0.0159    3.0000\r\n -289.0000   62.0000   71.0000    0.7193    1.0000\r\n  105.0000   42.0000    6.0000    0.5844    2.0000\r\n -342.0000    8.0000   38.0000    0.2064    2.0000\r\n   70.0000  -72.0000   33.0000    0.1787    4.0000\r\n -151.0000   35.0000   46.0000    0.5571    1.0000\r\n    3.0000  -70.0000   74.0000    0.5721    5.0000\r\n -177.0000  -12.0000   92.0000    0.2633    4.0000\r\n -103.0000   72.0000    3.0000    0.8341    1.0000\r\n -271.0000  -11.0000    4.0000    0.2545    1.0000\r\n  179.0000   76.0000   97.0000    0.1473    5.0000\r\n -114.0000   -6.0000    4.0000    0.5862    2.0000\r\n   45.0000   73.0000   32.0000    0.0335    5.0000\r\n  337.0000  -42.0000    4.0000    0.9355    5.0000\r\n  121.0000  -84.0000   37.0000    0.9568    4.0000\r\n  144.0000  -11.0000   85.0000    0.5869    3.0000\r\n -241.0000    3.0000    7.0000    0.4570    3.0000\r\n -343.0000  -77.0000   46.0000    0.9695    2.0000\r\n  258.0000  -60.0000   52.0000    0.6969    5.0000\r\n  -47.0000  -74.0000   48.0000    0.2568    3.0000\r\n -212.0000  -26.0000   38.0000    0.4174    5.0000\r\n -335.0000  -12.0000   69.0000    0.0045    3.0000\r\n  -33.0000  -66.0000   48.0000    0.5825    3.0000\r\n  161.0000  -85.0000   99.0000    0.4631    4.0000\r\n  218.0000   -7.0000   34.0000    0.4395    5.0000\r\n   57.0000  -74.0000   99.0000    0.8935    1.0000\r\n   -8.0000  -48.0000   66.0000    0.7985    3.0000\r\n  -98.0000   30.0000   69.0000    0.2574    2.0000\r\n  173.0000   23.0000   86.0000    0.8359    2.0000\r\n  186.0000  -49.0000   52.0000    0.0849    2.0000\r\n  -68.0000  -50.0000    9.0000    0.0143    4.0000\r\n  202.0000  -83.0000   88.0000    0.7898    3.0000\r\n -179.0000  -79.0000   11.0000    0.5538    5.0000\r\n  -76.0000   10.0000   67.0000    0.8318    4.0000\r\n  266.0000  -83.0000   49.0000    0.0301    4.0000\r\n -348.0000   80.0000   70.0000    0.0598    2.0000\r\n  309.0000   49.0000   62.0000    0.4251    1.0000\r\n  296.0000   -6.0000   20.0000    0.3192    5.0000\r\n  266.0000   77.0000   59.0000    0.8861    5.0000\r\n  144.0000  -77.0000   12.0000    0.6542    4.0000\r\n  -64.0000   54.0000   21.0000    0.0737    5.0000\r\n   71.0000  -79.0000   93.0000    0.9238    5.0000\r\n -170.0000  -34.0000   29.0000    0.5234    4.0000\r\n  111.0000   32.0000   88.0000    0.8344    3.0000\r\n   11.0000  -59.0000  100.0000    0.0446    1.0000\r\n  -35.0000   70.0000   71.0000    0.0496    3.0000\r\n  207.0000   52.0000   10.0000    0.5526    1.0000\r\n  209.0000  -89.0000   89.0000    0.5076    5.0000\r\n  309.0000   -2.0000   42.0000    0.1053    5.0000\r\n  353.0000  -62.0000   58.0000    0.6336    3.0000\r\n -191.0000  -59.0000   29.0000    0.1629    4.0000\r\n -262.0000         0   34.0000    0.5343    4.0000\r\n   28.0000  -47.0000   98.0000    0.7263    5.0000\r\n  328.0000   21.0000   43.0000    0.1479    3.0000\r\n -341.0000    8.0000   91.0000    0.5571    3.0000\r\n -324.0000  -60.0000   92.0000    0.1671    4.0000\r\n  -12.0000  -24.0000   83.0000    0.8690    2.0000\r\n   91.0000   30.0000   61.0000    0.4496    3.0000\r\n  230.0000  -20.0000   16.0000    0.3626    1.0000\r\n  128.0000   65.0000   82.0000    0.0067    1.0000\r\n   14.0000  -89.0000   61.0000    0.1293    4.0000\r\n  330.0000   39.0000   55.0000    0.0746    1.0000\r\n   71.0000   25.0000   27.0000    0.4566    2.0000\r\n  -32.0000   50.0000   74.0000    0.3240    1.0000\r\n  118.0000  -50.0000   54.0000    0.6298    5.0000\r\n  -55.0000   71.0000   46.0000    0.3064    3.0000\r\n -297.0000   47.0000   51.0000    0.8419    5.0000\r\n -114.0000   62.0000   42.0000    0.3906    3.0000\r\n   31.0000   69.0000   16.0000    0.1525    4.0000\r\n -284.0000   61.0000   18.0000    0.9337    4.0000\r\n  287.0000  -56.0000   38.0000    0.7202    5.0000\r\n  -43.0000   57.0000   11.0000    0.2651    4.0000\r\n -108.0000  -34.0000   99.0000    0.3242    4.0000\r\n   47.0000   -9.0000   21.0000    0.4719    1.0000\r\n  241.0000  -66.0000   36.0000    0.9853    4.0000\r\n  107.0000   55.0000   75.0000    0.5283    5.0000\r\n  -21.0000  -30.0000   69.0000    0.7108    3.0000\r\n -154.0000    8.0000   13.0000    0.9318    1.0000\r\n -182.0000   88.0000    1.0000    0.6157    2.0000\r\n  106.0000  -58.0000   23.0000    0.1413    5.0000\r\n  195.0000  -77.0000   52.0000    0.2124    5.0000\r\n -180.0000   50.0000   14.0000    0.8675    3.0000\r\n  -43.0000   36.0000   17.0000    0.2018    1.0000\r\n -147.0000  -80.0000   30.0000    0.1032    5.0000\r\n  -99.0000   36.0000   22.0000    0.4703    5.0000\r\n  198.0000    5.0000   24.0000    0.2535    4.0000\r\n   33.0000   -9.0000  100.0000    0.1854    5.0000\r\n  234.0000   18.0000   49.0000    0.3313    3.0000\r\n  200.0000   85.0000   26.0000    0.1587    5.0000\r\n  305.0000   75.0000  100.0000    0.4419    2.0000\r\n   36.0000   87.0000   67.0000    0.1831    5.0000\r\n  -50.0000   86.0000    7.0000    0.7323    4.0000];\r\ny_correct = [3.6203    3.4188  -11.9005];\r\nassert(max(abs(sphereCOM(x)-y_correct))\u003c1e-3)\r\n%%\r\n%Cheating is bad\r\nfunctions={'!','feval','eval','str2func','str2num','regex','system','dos','unix','perl','assert','fopen','write','save','setenv','path','please'};\r\nassessFunctionAbsence(functions, 'FileName', 'sphereCOM.m');","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":2002950,"edited_by":2002950,"edited_at":"2022-09-07T15:51:11.000Z","deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":"2022-09-07T15:51:11.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2022-03-03T14:59:29.000Z","updated_at":"2022-09-07T15:51:49.000Z","published_at":"2022-03-03T14:59:29.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach sphere has a position determined by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etheta\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (x,y plane angle) and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etau\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (elevation angle) as well as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, the distance of the center of the sphere from the origin. Each sphere also has a radius, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and a density of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003erho\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThese values are defined in a single input matrix: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esceneAttributes\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe output should be a 1x3 matrix defining the CoM in cartesian 3D space (x, y, z)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll angles are in degrees, all distances are in meters, and density is in kg/m^3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume density and lengths are always positive\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":49652,"title":"Find the spot diameter from the intensity distribution matrix of single spot (circle)","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 445.903px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.493px 222.951px; transform-origin: 406.493px 222.951px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIntensity distribution will be same as \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Normal_distribution\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egaussian dsitribution\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and check for \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003egaussian beam distribution\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 40px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 20px; text-align: left; transform-origin: 383.498px 20px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDistance between each index will be equal to 1 mm. Grid will be n*n matrix of distribution . n will be odd number . If n=3 center of the spot will be (2,2)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eFind spot diameter in mm.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eHint : w(z) is the radius of the laser beam where the irradiance is 1/e2 (13.5%) of Intensity\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eexample \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ex    0.0000    0.0045    0.0335    0.0045    0.0000\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0335   13.5335  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e100.0000 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  13.5335    0.0335\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    0.0000    0.0045    0.0335    0.0045    0.0000]\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI=100 (max intensity);\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI(13.5 %)=13.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eradius=1 mm;Spot diameter=2 mm\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.49px 10px; text-align: left; transform-origin: 383.498px 10px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Int(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nfiletext = fileread('Int.m');\r\nassert(isempty(strfind(filetext, 'regexp')),'regexp hacks are forbidden')\r\nassert(isempty(strfind(filetext, 'if')),'if is forbidden')\r\nassert(isempty(strfind(filetext, 'str')),'str is forbidden')\r\nassert(isempty(strfind(filetext, 'while')),'while is forbidden')\r\nassert(isempty(strfind(filetext, 'switch')),'switch is forbidden')\r\n\r\n\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0002    0.0045    0.0123    0.0045    0.0002    0.0000    0.0000\r\n    0.0000    0.0002    0.0335    0.6738    1.8316    0.6738    0.0335    0.0002    0.0000\r\n    0.0000    0.0045    0.6738   13.5335   36.7879   13.5335    0.6738    0.0045    0.0000\r\n    0.0000    0.0123    1.8316   36.7879  100.0000   36.7879    1.8316    0.0123    0.0000\r\n    0.0000    0.0045    0.6738   13.5335   36.7879   13.5335    0.6738    0.0045    0.0000\r\n    0.0000    0.0002    0.0335    0.6738    1.8316    0.6738    0.0335    0.0002    0.0000\r\n    0.0000    0.0000    0.0002    0.0045    0.0123    0.0045    0.0002    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),4))\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0003    0.0069    0.0189    0.0069    0.0003    0.0000    0.0000\r\n    0.0000    0.0003    0.0513    1.0309    2.8023    1.0309    0.0513    0.0003    0.0000\r\n    0.0000    0.0069    1.0309   20.7063   56.2856   20.7063    1.0309    0.0069    0.0000\r\n    0.0000    0.0189    2.8023   56.2856  153.0000   56.2856    2.8023    0.0189    0.0000\r\n    0.0000    0.0069    1.0309   20.7063   56.2856   20.7063    1.0309    0.0069    0.0000\r\n    0.0000    0.0003    0.0513    1.0309    2.8023    1.0309    0.0513    0.0003    0.0000\r\n    0.0000    0.0000    0.0003    0.0069    0.0189    0.0069    0.0003    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),4))\r\n\r\n%%\r\nx=[ 0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0003    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0188    0.1387    0.0188    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0003    0.1387    1.0250    0.1387    0.0003    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0188    0.1387    0.0188    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0003    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),2))\r\n\r\n%%\r\nx = [    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0005    0.0007    0.0005    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0091    0.0407    0.0671    0.0407    0.0091    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0247    0.3007    1.3476    2.2218    1.3476    0.3007    0.0247    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0091    0.3007    3.6631   16.4170   27.0671   16.4170    3.6631    0.3007    0.0091    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0005    0.0407    1.3476   16.4170   73.5759  121.3061   73.5759   16.4170    1.3476    0.0407    0.0005    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0007    0.0671    2.2218   27.0671  121.3061  200.0000  121.3061   27.0671    2.2218    0.0671    0.0007    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0005    0.0407    1.3476   16.4170   73.5759  121.3061   73.5759   16.4170    1.3476    0.0407    0.0005    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0091    0.3007    3.6631   16.4170   27.0671   16.4170    3.6631    0.3007    0.0091    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0247    0.3007    1.3476    2.2218    1.3476    0.3007    0.0247    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0007    0.0091    0.0407    0.0671    0.0407    0.0091    0.0007    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0005    0.0007    0.0005    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),8))\r\n%%\r\nx=[    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0036    0.0070    0.0036    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0018    0.0517    0.3818    0.7436    0.3818    0.0517    0.0018    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0517    1.4484   10.7022   20.8450   10.7022    1.4484    0.0517    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0036    0.3818   10.7022   79.0791  154.0251   79.0791   10.7022    0.3818    0.0036    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0070    0.7436   20.8450  154.0251  300.0000  154.0251   20.8450    0.7436    0.0070    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0036    0.3818   10.7022   79.0791  154.0251   79.0791   10.7022    0.3818    0.0036    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0517    1.4484   10.7022   20.8450   10.7022    1.4484    0.0517    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0018    0.0517    0.3818    0.7436    0.3818    0.0517    0.0018    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0005    0.0036    0.0070    0.0036    0.0005    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000]\r\n\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[   0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0002    0.0018    0.0035    0.0018    0.0002    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0009    0.0262    0.1934    0.3768    0.1934    0.0262    0.0009    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0002    0.0262    0.7338    5.4224   10.5615    5.4224    0.7338    0.0262    0.0002    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0018    0.1934    5.4224   40.0668   78.0394   40.0668    5.4224    0.1934    0.0018    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0035    0.3768   10.5615   78.0394  152.0000   78.0394   10.5615    0.3768    0.0035    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0018    0.1934    5.4224   40.0668   78.0394   40.0668    5.4224    0.1934    0.0018    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0002    0.0262    0.7338    5.4224   10.5615    5.4224    0.7338    0.0262    0.0002    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0009    0.0262    0.1934    0.3768    0.1934    0.0262    0.0009    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0002    0.0018    0.0035    0.0018    0.0002    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[ 0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0016    0.0120    0.0233    0.0120    0.0016    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0001    0.0061    0.1721    1.2714    2.4763    1.2714    0.1721    0.0061    0.0001    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0016    0.1721    4.8231   35.6383   69.4140   35.6383    4.8231    0.1721    0.0016    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0120    1.2714   35.6383  263.3335  512.9037  263.3335   35.6383    1.2714    0.0120    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0001    0.0233    2.4763   69.4140  512.9037  999.0000  512.9037   69.4140    2.4763    0.0233    0.0001    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0120    1.2714   35.6383  263.3335  512.9037  263.3335   35.6383    1.2714    0.0120    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0016    0.1721    4.8231   35.6383   69.4140   35.6383    4.8231    0.1721    0.0016    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0001    0.0061    0.1721    1.2714    2.4763    1.2714    0.1721    0.0061    0.0001    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0001    0.0016    0.0120    0.0233    0.0120    0.0016    0.0001    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0001    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000\r\n    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000    0.0000];\r\n\r\nassert(isequal(Int(x),6))\r\n\r\n%%\r\nx=[    0.0001    0.0007    0.0044    0.0197    0.0633    0.1456    0.2401    0.2837    0.2401    0.1456    0.0633    0.0197    0.0044    0.0007    0.0001\r\n    0.0007    0.0061    0.0384    0.1721    0.5525    1.2714    2.0961    2.4763    2.0961    1.2714    0.5525    0.1721    0.0384    0.0061    0.0007\r\n    0.0044    0.0384    0.2401    1.0762    3.4559    7.9520   13.1106   15.4883   13.1106    7.9520    3.4559    1.0762    0.2401    0.0384    0.0044\r\n    0.0197    0.1721    1.0762    4.8231   15.4883   35.6383   58.7577   69.4140   58.7577   35.6383   15.4883    4.8231    1.0762    0.1721    0.0197\r\n    0.0633    0.5525    3.4559   15.4883   49.7373  114.4443  188.6867  222.9070  188.6867  114.4443   49.7373   15.4883    3.4559    0.5525    0.0633\r\n    0.1456    1.2714    7.9520   35.6383  114.4443  263.3335  434.1636  512.9037  434.1636  263.3335  114.4443   35.6383    7.9520    1.2714    0.1456\r\n    0.2401    2.0961   13.1106   58.7577  188.6867  434.1636  715.8148  845.6352  715.8148  434.1636  188.6867   58.7577   13.1106    2.0961    0.2401\r\n    0.2837    2.4763   15.4883   69.4140  222.9070  512.9037  845.6352  999.0000  845.6352  512.9037  222.9070   69.4140   15.4883    2.4763    0.2837\r\n    0.2401    2.0961   13.1106   58.7577  188.6867  434.1636  715.8148  845.6352  715.8148  434.1636  188.6867   58.7577   13.1106    2.0961    0.2401\r\n    0.1456    1.2714    7.9520   35.6383  114.4443  263.3335  434.1636  512.9037  434.1636  263.3335  114.4443   35.6383    7.9520    1.2714    0.1456\r\n    0.0633    0.5525    3.4559   15.4883   49.7373  114.4443  188.6867  222.9070  188.6867  114.4443   49.7373   15.4883    3.4559    0.5525    0.0633\r\n    0.0197    0.1721    1.0762    4.8231   15.4883   35.6383   58.7577   69.4140   58.7577   35.6383   15.4883    4.8231    1.0762    0.1721    0.0197\r\n    0.0044    0.0384    0.2401    1.0762    3.4559    7.9520   13.1106   15.4883   13.1106    7.9520    3.4559    1.0762    0.2401    0.0384    0.0044\r\n    0.0007    0.0061    0.0384    0.1721    0.5525    1.2714    2.0961    2.4763    2.0961    1.2714    0.5525    0.1721    0.0384    0.0061    0.0007\r\n    0.0001    0.0007    0.0044    0.0197    0.0633    0.1456    0.2401    0.2837    0.2401    0.1456    0.0633    0.0197    0.0044    0.0007    0.0001];\r\n\r\nassert(isequal(Int(x),24))\r\n\r\n%%\r\nx=[  0.0000    0.0001    0.0005    0.0023    0.0077    0.0208    0.0454    0.0790    0.1103    0.1233    0.1103    0.0790    0.0454    0.0208    0.0077    0.0023    0.0005    0.0001    0.0000\r\n    0.0001    0.0007    0.0035    0.0149    0.0507    0.1378    0.2999    0.5227    0.7294    0.8152    0.7294    0.5227    0.2999    0.1378    0.0507    0.0149    0.0035    0.0007    0.0001\r\n    0.0005    0.0035    0.0186    0.0790    0.2683    0.7294    1.5877    2.7673    3.8621    4.3159    3.8621    2.7673    1.5877    0.7294    0.2683    0.0790    0.0186    0.0035    0.0005\r\n    0.0023    0.0149    0.0790    0.3351    1.1377    3.0925    6.7312   11.7319   16.3732   18.2973   16.3732   11.7319    6.7312    3.0925    1.1377    0.3351    0.0790    0.0149    0.0023\r\n    0.0077    0.0507    0.2683    1.1377    3.8621   10.4982   22.8506   39.8265   55.5824   62.1143   55.5824   39.8265   22.8506   10.4982    3.8621    1.1377    0.2683    0.0507    0.0077\r\n    0.0208    0.1378    0.7294    3.0925   10.4982   28.5369   62.1143  108.2597  151.0885  168.8443  151.0885  108.2597   62.1143   28.5369   10.4982    3.0925    0.7294    0.1378    0.0208\r\n    0.0454    0.2999    1.5877    6.7312   22.8506   62.1143  135.1999  235.6412  328.8638  367.5116  328.8638  235.6412  135.1999   62.1143   22.8506    6.7312    1.5877    0.2999    0.0454\r\n    0.0790    0.5227    2.7673   11.7319   39.8265  108.2597  235.6412  410.7012  573.1797  640.5392  573.1797  410.7012  235.6412  108.2597   39.8265   11.7319    2.7673    0.5227    0.0790\r\n    0.1103    0.7294    3.8621   16.3732   55.5824  151.0885  328.8638  573.1797  799.9367  893.9445  799.9367  573.1797  328.8638  151.0885   55.5824   16.3732    3.8621    0.7294    0.1103\r\n    0.1233    0.8152    4.3159   18.2973   62.1143  168.8443  367.5116  640.5392  893.9445  999.0000  893.9445  640.5392  367.5116  168.8443   62.1143   18.2973    4.3159    0.8152    0.1233\r\n    0.1103    0.7294    3.8621   16.3732   55.5824  151.0885  328.8638  573.1797  799.9367  893.9445  799.9367  573.1797  328.8638  151.0885   55.5824   16.3732    3.8621    0.7294    0.1103\r\n    0.0790    0.5227    2.7673   11.7319   39.8265  108.2597  235.6412  410.7012  573.1797  640.5392  573.1797  410.7012  235.6412  108.2597   39.8265   11.7319    2.7673    0.5227    0.0790\r\n    0.0454    0.2999    1.5877    6.7312   22.8506   62.1143  135.1999  235.6412  328.8638  367.5116  328.8638  235.6412  135.1999   62.1143   22.8506    6.7312    1.5877    0.2999    0.0454\r\n    0.0208    0.1378    0.7294    3.0925   10.4982   28.5369   62.1143  108.2597  151.0885  168.8443  151.0885  108.2597   62.1143   28.5369   10.4982    3.0925    0.7294    0.1378    0.0208\r\n    0.0077    0.0507    0.2683    1.1377    3.8621   10.4982   22.8506   39.8265   55.5824   62.1143   55.5824   39.8265   22.8506   10.4982    3.8621    1.1377    0.2683    0.0507    0.0077\r\n    0.0023    0.0149    0.0790    0.3351    1.1377    3.0925    6.7312   11.7319   16.3732   18.2973   16.3732   11.7319    6.7312    3.0925    1.1377    0.3351    0.0790    0.0149    0.0023\r\n    0.0005    0.0035    0.0186    0.0790    0.2683    0.7294    1.5877    2.7673    3.8621    4.3159    3.8621    2.7673    1.5877    0.7294    0.2683    0.0790    0.0186    0.0035    0.0005\r\n    0.0001    0.0007    0.0035    0.0149    0.0507    0.1378    0.2999    0.5227    0.7294    0.8152    0.7294    0.5227    0.2999    0.1378    0.0507    0.0149    0.0035    0.0007    0.0001\r\n    0.0000    0.0001    0.0005    0.0023    0.0077    0.0208    0.0454    0.0790    0.1103    0.1233    0.1103    0.0790    0.0454    0.0208    0.0077    0.0023    0.0005    0.0001    0.0000];\r\n\r\nassert(isequal(Int(x),36))","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":610936,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2021-01-05T18:44:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2020-12-28T20:35:54.000Z","updated_at":"2025-12-09T19:47:07.000Z","published_at":"2020-12-28T21:31:27.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIntensity distribution will be same as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Normal_distribution\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egaussian dsitribution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and check for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.edmundoptics.com/knowledge-center/application-notes/lasers/gaussian-beam-propagation/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003egaussian beam distribution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDistance between each index will be equal to 1 mm. Grid will be n*n matrix of distribution . n will be odd number . If n=3 center of the spot will be (2,2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFind spot diameter in mm.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eHint : w(z) is the radius of the laser beam where the irradiance is 1/e2 (13.5%) of Intensity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eexample \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex    0.0000    0.0045    0.0335    0.0045    0.0000\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0335   13.5335  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e100.0000 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  13.5335    0.0335\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0045    1.8316   13.5335    1.8316    0.0045\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    0.0000    0.0045    0.0335    0.0045    0.0000]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI=100 (max intensity);\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI(13.5 %)=13.5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eradius=1 mm;Spot diameter=2 mm\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":44741,"title":"You are constantly moving at a speed v faster than your twin brother. How long does it take before you become 1s younger than him according to the theory of relativity?","description":"You are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\r\n\r\nDefine:\r\ngam = 1/sqrt(1-v^2/c^2)   where v is the relative speed in m/s and c is the speed of light in m/s.\r\n\r\nAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?","description_html":"\u003cp\u003eYou are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\u003c/p\u003e\u003cp\u003eDefine:\r\ngam = 1/sqrt(1-v^2/c^2)   where v is the relative speed in m/s and c is the speed of light in m/s.\u003c/p\u003e\u003cp\u003eAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?\u003c/p\u003e","function_template":"function years = becomeOneSecondYounger(v_in)\r\n    \r\nend","test_suite":"%%\r\nv_in = 100;            % km/h\r\nyearsCorrect = 241830; % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c10)\r\n\r\n\r\n%%\r\nv_in = 1000;           % km/h\r\nyearsCorrect = 2418.7; % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c0.1)\r\n\r\n%%\r\nv_in = 10000;           % km/h\r\nyearsCorrect = 24.187;  % years\r\nyears = becomeOneSecondYounger(v_in);\r\nassert(abs(yearsCorrect-years)\u003c0.01)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":195293,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":4,"test_suite_updated_at":"2018-10-03T10:36:13.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2018-10-03T08:07:37.000Z","updated_at":"2018-10-03T10:36:13.000Z","published_at":"2018-10-03T10:32:42.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou are moving at a speed of v_in (km/h) relative to your twin brother of the exact (even in seconds) same age.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefine: gam = 1/sqrt(1-v^2/c^2) where v is the relative speed in m/s and c is the speed of light in m/s.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAccording to the theory of relativity (specifically time dilation), a time interval dt for your twin brother equals a time interval of gam*dt for you. For how long must you travel, in years, with the speed v_in in order to become 1 second younger than your twin brother?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2316,"title":"Spin Matrices","description":"The spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\r\nSuppose you have integer or half-integer spin of value s. The matrices Sx, Sy and Sz representing it have the following properties:\r\nSi (with i={x,y,z}) are traceless Hermitian matrices;\r\nCommutation relations (a): [ Si,Sj ] = i εijk Sk, where [·,·] is the commutator and εijk is the Levi-Civita symbol.\r\nCommutation relations (b): [ Si,S² ] = 0, where S² = Sx²+Sy²+Sz²;\r\nEigenvalues: S² = j(j+1)·I and Sz = diag( -j/2, -j/2+1, … ,j/2-1, j ), where I is the identity matrix.\r\nSee also this article for more reference.\r\nExamples\r\n [Sx,Sy,Sz] = spin_matrices(1/2)\r\n\r\n Sx = \r\n     0      0.5\r\n     0.5    0\r\n\r\n Sy = \r\n     0     -0.5i\r\n     0.5i   0\r\n\r\n Sz = \r\n     0.5    0\r\n     0     -0.5\r\nNote:\r\nThe usual cheats are not allowed!","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 592.367px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 296.183px; transform-origin: 407px 296.183px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 384px 8px; transform-origin: 384px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 171.5px 8px; transform-origin: 171.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSuppose you have integer or half-integer spin of value\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.5px 8px; transform-origin: 3.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003es\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45px 8px; transform-origin: 45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The matrices\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.5px 8px; transform-origin: 7.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSx\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSy\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSz\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 108px 8px; transform-origin: 108px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e representing it have the following properties:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 81.7333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 40.8667px; transform-origin: 391px 40.8667px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 16.5px 8px; transform-origin: 16.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e (with\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ei={x,y,z}\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15px 8px; transform-origin: 15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e) are\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etraceless\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eHermitian matrices\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 87.5px 8px; transform-origin: 87.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCommutation relations (a): [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi,Sj\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 13.5px 8px; transform-origin: 13.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ] = i\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eεijk Sk\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 55px 8px; transform-origin: 55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where [·,·] is the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ecommutator\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10px 8px; transform-origin: 10px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eεijk\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19px 8px; transform-origin: 19px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eLevi-Civita symbol\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 87.5px 8px; transform-origin: 87.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCommutation relations (b): [\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15px 8px; transform-origin: 15px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSi,S²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39px 8px; transform-origin: 39px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ] = 0, where\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eS²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e =\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39px 8px; transform-origin: 39px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSx²+Sy²+Sz²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 40.5px 8px; transform-origin: 40.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEigenvalues:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7px 8px; transform-origin: 7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eS²\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e =\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 20.5px 8px; transform-origin: 20.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ej(j+1)·I\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 8px 8px; transform-origin: 8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eSz\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 24px 8px; transform-origin: 24px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e = diag(\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66.5px 8px; transform-origin: 66.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e-j/2, -j/2+1, … ,j/2-1, j\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 27.5px 8px; transform-origin: 27.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ), where\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eI\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 66px 8px; transform-origin: 66px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the identity matrix.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.5px 8px; transform-origin: 27.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ethis article\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63px 8px; transform-origin: 63px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for more reference.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 33.5px 8px; transform-origin: 33.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 265.633px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 132.817px; transform-origin: 404px 132.817px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 128px 8.5px; tab-size: 4; transform-origin: 128px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e [Sx,Sy,Sz] = spin_matrices(1/2)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sx = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0      0.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5    0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sy = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 64px 8.5px; tab-size: 4; transform-origin: 64px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     -0.5i\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5i   0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 8.5px; tab-size: 4; transform-origin: 0px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 24px 8.5px; tab-size: 4; transform-origin: 24px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e Sz = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 52px 8.5px; tab-size: 4; transform-origin: 52px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0.5    0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 60px 8.5px; tab-size: 4; transform-origin: 60px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     -0.5\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18.5px 8px; transform-origin: 18.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNote:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 54.5px 8px; transform-origin: 54.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe usual cheats\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eare not\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e allowed!\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [Sx,Sy,Sz] = spin_matrices(s)\r\n  [Sx,Sy,Sz] = deal(s);\r\nend","test_suite":"%%\r\nuser_solution = fileread('spin_matrices.m');\r\nassert(isempty(strfind(user_solution,'regexp')));\r\nassert(isempty(strfind(user_solution,'num2str')));\r\nassert(isempty(strfind(user_solution,'interp')));\r\nassert(isempty(strfind(user_solution,'fprintf')));\r\nassert(isempty(strfind(user_solution,'assert')));\r\n\r\n%%\r\n% use auxiliary functions\r\niseq = @(x,y)norm(x-y)\u003c=64*eps; % check if equal up to 64 eps\r\ncom  = @(x,y)x*y-y*x;           % commutator\r\ntrace= @(x)  sum(diag(x));      % trace\r\n%\r\nfprintf('Testing...\\n')\r\nfor s = 1/2:1/2:5,\r\n   % Get the matrices\r\n   fprintf('\\ts=%-3s : ',strtrim(rats(s)));\r\n   [Sx,Sy,Sz] = spin_matrices(s);\r\n   %\r\n   % ancillary parameters\r\n   mz   = (-s:s)';                            % eigenvalues\r\n   S2   = Sx^2+Sy^2+Sz^2;                     % S^2 matrix\r\n   %\r\n   assert(trace(Sx)==0\u0026\u0026iseq(Sx,Sx'),'Sx must be a traceless Hermitian matrix');\r\n   assert(trace(Sy)==0\u0026\u0026iseq(Sy,Sy'),'Sy must be a traceless Hermitian matrix');\r\n   assert(trace(Sz)==0\u0026\u0026iseq(Sz,Sz'),'Sz must be a traceless Hermitian matrix');\r\n   %\r\n   % actual values\r\n   assert(iseq(com(Sx,Sy),1i*Sz), 'Commutation relations: [Sx,Sy] = i*Sz')\r\n   assert(iseq(com(Sy,Sz),1i*Sx), 'Commutation relations: [Sy,Sz] = i*Sx')\r\n   assert(iseq(com(Sz,Sx),1i*Sy), 'Commutation relations: [Sz,Sx] = i*Sy')\r\n   %\r\n   assert(iseq(S2,s*(s+1)*eye(2*s+1)), 'S^2 must be a quantum number!');\r\n   assert(iseq(eig(Sz),mz),            'Sz must be a quantum number!');\r\n   %\r\n   fprintf('OK!\\n');\r\nend\r\n%\r\nfprintf('\\n \\nWolfgang Pauli would be proud!\\n')\r\n%","published":true,"deleted":false,"likes_count":3,"comments_count":3,"created_by":10352,"edited_by":223089,"edited_at":"2022-08-20T10:34:48.000Z","deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":"2014-05-11T14:57:22.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-05-11T11:28:29.000Z","updated_at":"2024-11-18T01:26:43.000Z","published_at":"2014-05-11T14:54:32.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose you have integer or half-integer spin of value\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The matrices\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSx\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSy\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSz\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e representing it have the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e (with\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ei={x,y,z}\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e) are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etraceless\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eHermitian matrices\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCommutation relations (a): [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi,Sj\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ] = i\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eεijk Sk\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, where [·,·] is the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ecommutator\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eεijk\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eLevi-Civita symbol\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCommutation relations (b): [\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSi,S²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ] = 0, where\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e =\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSx²+Sy²+Sz²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEigenvalues:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eS²\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e =\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ej(j+1)·I\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSz\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e = diag(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e-j/2, -j/2+1, … ,j/2-1, j\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e ), where\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the identity matrix.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ethis article\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for more reference.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ [Sx,Sy,Sz] = spin_matrices(1/2)\\n\\n Sx = \\n     0      0.5\\n     0.5    0\\n\\n Sy = \\n     0     -0.5i\\n     0.5i   0\\n\\n Sz = \\n     0.5    0\\n     0     -0.5]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNote:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe usual cheats\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare not\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e allowed!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":57770,"title":"Number of images formed due to two inclined mirrors and their coordinates","description":"There are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. Final output should contain a vector of coordinates including the original object coordinate.\r\n ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 249px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 124.5px; transform-origin: 407px 124.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThere are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. Final output should contain a vector of coordinates including the original object coordinate.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 156px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 78px; text-align: left; transform-origin: 384px 78px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cimg class=\"imageNode\" width=\"170\" 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data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [N,y] = images(x,theta)\r\n  \r\nend","test_suite":"%%\r\nx = 2;\r\ntheta = pi;\r\nN_correct = 1;\r\ny_correct = [2 -2;0 0];\r\n[N,y] = images(x,theta)\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n\r\n%%\r\nx = 20;\r\ntheta = pi/4;\r\nN_correct = 7;\r\ny_correct =   [20    14     0   -14   -20   -14     0    14;\r\n                0    14    20    14     0   -14   -20   -14]\r\n\r\n[N,y] = images(x,theta)\r\n    \r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%% \r\nx = 1.1\r\ntheta = pi/2;\r\nN_correct = 3;\r\ny_correct =   [1.1000         0   -1.0000         0;\r\n                    0    1.0000         0   -1.0000];\r\n[N,y] = images(x,theta)\r\n\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%%\r\nx = -10;\r\ntheta = 2*pi;\r\nN_correct = 0;\r\ny_correct = [-10;0];\r\n[N,y] = images(x,theta)\r\n\r\nassert(isequal(y,y_correct))\r\nassert(isequal(N,N_correct))\r\n\r\n%%\r\nx = 20;\r\ntheta = 0;\r\nN_correct = Inf;\r\n[N,y] = images(x,theta);\r\n\r\nassert(isequal(N,N_correct))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":1809490,"edited_by":1809490,"edited_at":"2023-03-12T09:24:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":"2023-03-12T09:24:24.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2023-03-12T08:45:13.000Z","updated_at":"2023-03-12T09:24:24.000Z","published_at":"2023-03-12T09:10:51.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are two mirrors with angle theta between them and there is an object on x axis. Determine the number of images formed and the cordinates of the images when coordinate of the object on x axis is given. The coordinates will be given as x = 2 or x = 3 etc. Round the coordinates to the nearest integers. 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Partial Differential Equations: Explicit Method","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 294px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 147px; transform-origin: 407px 147px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eI got this example from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003enumericalmethodsguy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e  youtube video. He solves the problem step by step.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://www.youtube.com/watch?v=p0V1eSlM2xo\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eOn the second test: I varied the length of rod and the final time. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efrom t = 0 and t = 102 seconds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function T = pderod(L,tf)\r\n% defined constants\r\nk=54;                %W/(m*K)\r\np=7800;              %kg/m^3\r\nC=490;               %J/(kg*K)\r\nL0 = 0;               %meter \r\nnode = 5;            % segments\r\ndt = 3;              %second time steps\r\nt0=0;                %s initial time\r\nT0 = 100;           %C initial Temperature   \r\nTend = 25;          %C final Temperature\r\nTm1 =20;            %C\r\nTm = repelem(Tm1,node-1)'; %C middle Temperature \r\nTi=[T0;Tm;Tend];     % Celsius at Time 0second \r\nT0=Ti;               % Celsius at Time 0second \r\n\r\nend","test_suite":"%%\r\n% INPUTS\r\nL = .05;                     %meter\r\ntf = 9;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   53.9089   59.0725   65.9508   85.0000\r\n   20.0000   20.0000   34.3727   39.1307   70.0000\r\n   20.0000   20.0000   20.8983   27.2639   55.0000\r\n   20.0000   22.1193   22.4420   22.8719   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend\r\n%%\r\n% INPUTS\r\nL = .10;                     %meter\r\ntf = 102;                      %s final time\r\nT = pderod(L,tf)             % C\r\nT_correct = [100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000  100.0000\r\n   20.0000   28.4772   35.1579   40.5179   44.8926   48.5211   51.5762   54.1838   56.4375   58.4071   60.1460   61.6949   63.0856   64.3431   65.4874   66.5344   67.4973   68.3867   69.2115   69.9791   70.6957   71.3666   71.9963   72.5887   73.1472   73.6746   74.1735   74.6463   75.0947   75.5206   75.9256   76.3111   76.6782   77.0283   77.3622   85.0000\r\n   20.0000   20.0000   20.8983   22.3201   24.0280   25.8726   27.7607   29.6354   31.4629   33.2239   34.9089   36.5138   38.0384   39.4848   40.8559   42.1556   43.3879   44.5569   45.6666   46.7206   47.7226   48.6758   49.5832   50.4477   51.2717   52.0578   52.8081   53.5245   54.2090   54.8632   55.4887   56.0871   56.6596   57.2077   57.7323   70.0000\r\n   20.0000   20.0000   20.0561   20.2398   20.5707   21.0425   21.6372   22.3328   23.1075   23.9414   24.8173   25.7209   26.6402   27.5657   28.4898   29.4063   30.3106   31.1990   32.0688   32.9179   33.7447   34.5484   35.3281   36.0835   36.8145   37.5211   38.2035   38.8621   39.4971   40.1091   40.6987   41.2663   41.8126   42.3382   42.8437   55.0000\r\n   20.0000   20.5298   20.9474   21.2824   21.5658   21.8243   22.0780   22.3409   22.6218   22.9253   23.2528   23.6037   23.9760   24.3669   24.7729   25.1908   25.6173   26.0492   26.4837   26.9183   27.3508   27.7792   28.2020   28.6178   29.0256   29.4244   29.8135   30.1925   30.5609   30.9186   31.2653   31.6010   31.9257   32.2395   32.5425   40.0000\r\n   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000   25.0000];\r\n%ismembertol(A, B, 0.05, 'ByRows', true)\r\n[rows cols] = size(T);\r\nfor i =1:rows\r\nassert(ismembertol(T(i,:),T_correct,.2,'ByRows',true));\r\nend","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":227209,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-07-31T20:44:36.000Z","updated_at":"2020-08-17T00:46:40.000Z","published_at":"2020-07-31T21:36:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eI got this example from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/channel/UCtXs16H04R0SSeRI8UEXMxw\\\"\u003e\u003cw:r\u003e\u003cw:t\u003enumericalmethodsguy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e  youtube video. He solves the problem step by step.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.youtube.com/watch?v=p0V1eSlM2xo\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttps://www.youtube.com/watch?v=p0V1eSlM2xo\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA rod of steel is subjected to a temperature of 100°C on the left end and 25°C on the right end. If the rod is of length 0.05m, use the explicit method to find the temperature distribution in the rod from t = 0 and t = 9 seconds. Use ∆x = 0.01m , ∆t = 3s . Given:  k =-54 W/(m*K) ,  ρ = 7800 kg/m^3, C = 490 J/(kG*K) . The initial temperature of the rod is 20°C.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eMake a matrix of all Temperatures in °C. The first column is the initial conditions at time 0 second, the middles columns are the unknow tempertures, and the final column is the linear path from the intial conditions to the final conditions.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the second test: I varied the length of rod and the final time. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003erod is of length 0.10m  and ∆x = 0.02m  \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efrom t = 0 and t = 102 seconds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":1109,"title":"USC Spring 2012 ACM: Armageddon","description":"This Challenge is to solve Question E, Armageddon, of the \u003chttp://contest.usc.edu/index.php/Spring12/Home USC ACM Spring 2012 Contest\u003e.\r\n\r\nAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u003e0, \u003c180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km\r\n, and angle in degrees.\r\n\r\nReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places. \r\n\r\n\r\n*Input: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]*\r\n\r\n*Output: [Maximum Time Prior to Launch]*; if too late return -1;\r\n\r\n\r\nThe full \u003chttp://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=armageddon.in.txt USC data file\u003e\r\n\r\n*Example:*\r\n\r\n*Input: 63781.0 20.9514 6378.1 6378.1*\r\n\r\n*Output: 0.00* as immediate Launch is required  \r\n\r\nInput 47835.75,15,6000,5000 returns -1.\r\n\r\n\r\n\u003chttp://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=numerology_darryl.cpp.txt The Judges' E solution\u003e.\r\n\r\nGeometry Hint: Draw a circle and a line from the center of  approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\r\n","description_html":"\u003cp\u003eThis Challenge is to solve Question E, Armageddon, of the \u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home\"\u003eUSC ACM Spring 2012 Contest\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u003e0, \u0026lt;180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km\r\n, and angle in degrees.\u003c/p\u003e\u003cp\u003eReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput: [Maximum Time Prior to Launch]\u003c/b\u003e; if too late return -1;\u003c/p\u003e\u003cp\u003eThe full \u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=armageddon.in.txt\"\u003eUSC data file\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eExample:\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput: 63781.0 20.9514 6378.1 6378.1\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput: 0.00\u003c/b\u003e as immediate Launch is required\u003c/p\u003e\u003cp\u003eInput 47835.75,15,6000,5000 returns -1.\u003c/p\u003e\u003cp\u003e\u003ca href=\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=numerology_darryl.cpp.txt\"\u003eThe Judges' E solution\u003c/a\u003e.\u003c/p\u003e\u003cp\u003eGeometry Hint: Draw a circle and a line from the center of  approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\u003c/p\u003e","function_template":"function [t_launch]=Armageddon(xa,angle,va,vm)\r\n  t_launch=-1;\r\nend","test_suite":"%%\r\n% Armegeddon\r\ntic\r\nurlwrite('http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026upname=armageddon.in.txt','armageddon.in.txt')\r\ntoc\r\n%%\r\n fid=fopen('armageddon.in.txt','r');\r\n \r\n t_expect=[0.00 -1 20.55 -1 -1 28.38 -1 11.03 2.62 4.22 13.15 9.94 61.33 13.56 -1];\r\n \r\n  \r\n qty=fscanf(fid,'%i',1);\r\n for q=1:qty %qty\r\n  n = fscanf(fid,'%f %f %f %f \\n',4)'; % dist, angle, vel A, vel Missile\r\n  xa=n(1);\r\n  angle=n(2);\r\n  va=n(3);\r\n  vm=n(4);\r\n  \r\n  [t]=Armageddon(xa,angle,va,vm) ;\r\n  \r\n  \r\n  assert(isequal(t,t_expect(q)))\r\n  \r\n   \r\n  end\r\n   \r\n fclose(fid);\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":10,"test_suite_updated_at":"2012-12-09T06:02:07.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-12-08T21:43:00.000Z","updated_at":"2012-12-09T15:08:14.000Z","published_at":"2012-12-08T22:02:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to solve Question E, Armageddon, of the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eUSC ACM Spring 2012 Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn asteroid has been detected that will traverse the center of the earth if it is not deflected. The Splitter missile will bisect the asteroid with an included angle (\u0026gt;0, \u0026lt;180 degrees), based on its composition. The asteroids distance(to center of earth), velocity, and split angle will be provided. The Splitter missile velocity is also provided. Launch occurs from the earth surface nearest the asteroid. Miscellaneous details: Earth Radius 6378.1 km, velocities in km/sec, distance in km , and angle in degrees.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eReturn the Maximum Time Prior to Launch, nothing hitting the earth, to two decimal places.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput: [ Asteroid Distance, Angle of Split, Asteroid Velocity, Missile Velocity ]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput: [Maximum Time Prior to Launch]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; if too late return -1;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe full\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=armageddon.in.txt\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eUSC data file\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput: 63781.0 20.9514 6378.1 6378.1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput: 0.00\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e as immediate Launch is required\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInput 47835.75,15,6000,5000 returns -1.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://contest.usc.edu/index.php/Spring12/Home?action=download\u0026amp;upname=numerology_darryl.cpp.txt\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eThe Judges' E solution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGeometry Hint: Draw a circle and a line from the center of approximately 2 radius. Take a ruler pivoting from the end of the line and rotate the ruler until it touches the circle.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":46938,"title":"Numerical computation of the optimal shooting angle of a catapult","description":null,"description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 879.667px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 406.5px 439.833px; transform-origin: 406.5px 439.833px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 64.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 32.1667px; text-align: left; transform-origin: 383.5px 32.1667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a capapult that fires a projects into the air with an initial velocity\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANoAAAAqCAYAAADVj0XCAAAKM0lEQVR4Xu2cd6w2VRGHH+xdEewGW4gao2Cw94IaxR5jQTQC9oZdFLuCINixiwTsJYoSomAUNWILJmCJsYsl9hIVG7Y8+ebw7bfflrO7777svfecv27uu2XOnPmdmfnNnN2FMooGigZm18Aus7+hvKBooGiAArRiBEUDa9BAAdoalFxeUTRQgFZsoGhgDRooQFuDkssrigYK0LbZwGWAA4DLN5jE54GvbFFTuSrwUODiDfP/KPD9LaoXpy127gA8Ebgl8A/gFOB1wM/qeilA26aR3YD3AJcG/l5T0puBj29Rg7ou8ErgCpX5XwjYE3g4cMYW1YvTdgN6GvAW4Fzg0cA9wlYeB/y6qpsCtO1AOxZ4CfDdLWw8OVO/JPBa4N1bGGhXA14NvBz4TijNTVq9PCaio/cWoO1sTnq0ArQcmEEBGtwWuAHwLuB/FbXdNzzaiwKE5/9UPFrxaHnw2n7VRgOaoPhixiT/ApwF/BA4Ffg08PuW+y4W//9X7ff0rscDbyserXi0DLtrvWQjAs086mDgR8AzAEO/FwaBkSbqvG4HvCzIsT8DLwU+DPw7U2F6tLcC9we+VoBWgJZpN42XbUSgvQl4SOTfAuwawNMbiK/rAx8MMF4HOAI4BPhAhsIuDLwYuHLTs0voWELHDBva4ZKtALQnAV8FDgXuCDwS+GWPom4IHA08u0KQlBytprRChuTDbZVA81m3Bx4G3BrQo/QNw7shZQXzpqEeTaD5jpuENzuop5Yq4ygDaRnIuutOo3i04tH6DLv++yqApt3dGTgSuPlAAdYJNDdgaXpLGTvQ9RWZLxKh5s+B99dYyOLRikcbaN7bL58KNEFmvqSXuWIwfK8HvhCF39GCtdw4xaP1Ac256I2vCbymizRp82jGmyaCl+qYtZ0UIn0zjNzQ0RDBHfgB0QVwJeBRwMmhhMtGjG7oIT38FOC3MynIdynLTYGbRdh1NvDkmsFqaMcB3wh5duhYGCHbVKApj97hWtEg8KoGUmKEWK23TAGatbL3RZtVvQ1P7GgHkibWYP9ZkUD83Aj4SPpfG9BsrxFIbcO2pOfMsAP1vTd3AYaGF7lAc+dynBeJ7yNisxFY/412JQ3R3rc/RBL901yhB15nyCLYLgccFh0JJux1anm/CGm81rYhWbUpYwrQ3Kg0Sjcn2T9Dx1zqfKzMY4H2pWirenALGSJJYnfIJ4G/VoSzxmYPpO1Z3+4Cmhfqzb4ZC1QtyonUZ8bCTt0ZmxS3dKBVZbafzZrJ1wEBdx+gM04faykZ9107NkaNSgM+vJIr2Jt4zwjVbA/S004ZU4C2T3h/61k2cf9kiiCZ944FmhvZO8Lr1nOvvWPt3VCbxhvCEZ3v5Zo8muGQdKZeq9pge5VYQFGc+rsy57r4y3I9WnUiyWgsftoF8LtYlLl36CZlphqOIPtUAF950nBNzSFeAJwzcTWmAC1tTvXNYKJInbc3Ac1wr16w9iECyOK2xeZbAALGzXTymuayjr305ZyaWsOzxwDt6pFr3Ak4PYz7F2uQte0VdwU+FmGM3lVPm4ZGYwhkiFnNJcaIOwVoGredF1PG0LSgCWhtMvyGbb2c5mWGtW5K1V7G0XLnAC2LvqxJ4HPNHdw5bhy53EmRx4wWdsYbxwDtEhGjex7Jrm2Lm/XetzaR59BPFfjVXju9nU2usnqf6dBhrkybAWipM8SwWjJGj6+3/VOwiJJY1s5WFrn1AS2bvmwAmmeY3EH3jyTchV7qGAM0dWMXwFEt4VrXXL131foxt3YXtrXIUDa1GFkEdjN4fg95lSvTFKCl0NF+QzendYy+HE3QyZ7bSGxa5CFXwef/dzpXNlbgPqDJrNwv4lkPtw0Z7qRWy3X1gk2ioG9sJDJESl36Vprak8YPCgq9b47p9zH66Xt20t/n4mCmoZBn7Axtu7zZEJmmAO1uwGkR4trk+8e+Ca3g9z6g+Qrt3FqefY12dshHuFn9YEXhdufn5qYyjLvHTmGc29TA2aTDjQI0F8KFkcEzxLhN02G/HiMZo58+u0sEjXSzwJfStzj83MxaVY5MU4AmcXRilD+UT9DNPXKAZnokm25um9hQN1JzNTeq1o6PXOHbPNoqGEYTcPMyO5rfuaqkMndiA68bEjq6KNYQpabtbTPcMMTwyL/JvvU0+/c8qd1VAmnTj4Zs/uf5qKFsl7mGxrFv1Ko0srcDZ2bqI2fNpgBNe3tqfFfD72t4/P9XmbKNvSwHaD5bTuGNQSa5vn+LfE2gWRZp7GHMFaoJaGMZRu9zRzAnMM61uHurKBUs/eM2XUBLJMF/Isex5cZ5JS+R8o5Eq9sRcPdg16oMX45+jCKOjx3fplbDqyGfVqgSNBavzYOk9ZW9aeTIVL9vCtB8VgrLTEk862W0Mydbmws0ZVP/npoWcHoxw/u6p8vF1g7X1YGWXKhJtccETghX35ef+bGWY4BPROFUtvFDwI9jNzBXWPLoApq1FT2zfXnfA+z0eELFWyUvYFikhzNsc3Gq3ixXP/XQeUy9ScMVXH0eI1emVQMtGXTaUNxIJJTU8Rw5m0CrH/x0vZvSmUT+6cX0tnZ2+CUw0wTlHN0NVQVaeokFOmP7NKrtVu6AMkbWFgyZLGinXdhWlNS3Zo+YQLMH0NBxaAi0blB2Ac0d2Ll6etYDgFLl1XDHHd45WuTXq/l39XNjQ/RjNGCzrTUxR5U9zNWJYPVclLnGZ1tuGiLTHEDzmZJIelz7BYeMMXW0+qcMuvSqs/FktSxtfVjC0RYG23MVaLsGQ+jiaFyPDW/kTm7fllXy61VOm9o3p8sXXHsBB8b3FhQusUsPDIZpiCIviGuH5GhD5BOEY/UjcPeotVP1vdv1dKe+aIdBTJHJ908NHatz0KhtY7KY7qfa5jiP1qeztfzeR+/bQye6pT91m9Ya3IUMi3Sv7i52I2hMqUE0tQPpAdLx8bVMZsJL5gJa6tYYqh8jh1dEvmbXfe6QKXODfBbgNy+axliZ0rNWCbTceW346/qA5gS9RmLDBF+vJ4P15eiCsCAtAHX/qU5jfmYiaUxrhb1twZekvDmApt7G6GdMJ466zGGKx8pUXasCtBGWmwO0tscmhd+lUqyVbbRzXDLAcNOjNl6XU6weIf7KbhFo5qISOnaWV4c0b/3rxTkvHqMfWUPpb09OGD1YKmgarpvsptd7rR5QFtQOh291CDdUpnQUx1alNHynjQiecRvySYEcnW3aa6YALXU2PC+ApmFYWfc7C8bc0uB6N+s6UzvG516AdJLWPKE+JBXajrF3yTWnfgSZ1LikhqG7rLDffK82EjfJNlSm9FUoc/D6GEpKzL2Gi37+FKA5MSliqeR7R+honuaOZ5FUBs4u6XWcOVqqkufSTzr2cq/I4/TEuYXfuWRa6hosQq6pQFvEJIoQRQNL10AB2tJXqMi3KTRQgLYplrFMYukaKEBb+goV+TaFBgrQNsUylkksXQMFaEtfoSLfptDA/wF1oOpJsndYXQAAAABJRU5ErkJggg==\" width=\"109\" height=\"21\" style=\"width: 109px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and an initial velocity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"56.5\" height=\"20\" style=\"width: 56.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, find the optimal shooting angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"18.5\" style=\"width: 16px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eof the catapult that minimizes the distance between the target and the trajectory of the fired projectile. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 21.25px; text-align: left; transform-origin: 383.5px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etip 1:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e Consider the states \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the x- and y-position of the projectile, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45\" height=\"22\" style=\"width: 45px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,     \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"45.5\" height=\"22\" style=\"width: 45.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e      \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 11px; text-align: left; transform-origin: 383.5px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; 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height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"47.5\" height=\"17.5\" style=\"width: 47.5px; height: 17.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eis the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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Plotting \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e vs. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e will result in the x-y trajectory of the projectile, as shown in the figure below.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003etip 2: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eUse the following update law, to incrementally update the shooting angle \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"20\" style=\"width: 15px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.9167px; text-align: left; transform-origin: 383.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e        \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATMAAAAoCAYAAACSPh2yAAAR5UlEQVR4Xu3dB7A1S1UF4IUZIwgGzFYRFDFizmIExEgyoJhFzDlHzBkDKiKYURCzICbAgKAIiqKoZc4551Tfq97PZpg5M+eeOeem6aq//vf+O6Fnd/faa6+9u+9NsrXNApsFNgtcAQvc5Ap8w/YJmwU2C2wWyAZm2yTYLLBZ4EpYYAOzKzGM20dsFtgssIHZNgc2C2wWuBIW2MDsSgzj9hGX3AIvmuQ1k/xUkv+5wN/yskleOsmTkvzvRevnBmYXbUS2/lw3C9wmyYcl+dwkf37BP/45krx7kudO8q1J/usi9XdfMHuuJK+W5PWTvFCSX03yuCT/fJE+6pL25aZJXivJ6yZ5ziS/mOSJSf79kn7P1u15C9wuyUcn+fQ9gezF2xp85ST/mOQnkvzmidgSzLh3kue7aIC2FMxc9wZJviLJ45N8cZI7JPmmJE9L8kF7Dsb8MF+fK3i7d0nymUkeluTrk7xjkq9tk+Xjr5CzeLHGQN4+yRcm+eok/73yUD9vko9L8iFJPqXZ9MKFREnY4ivbeFtTSxoC8YlJ7pzkE5I8OcnnJ3nPJB+c5DtPBGgcLyb5Q0l+cknHT3HNEjBzzTsn+bIkP5ykFhfDfk2S+yT5yCQPPJEhT2GXU70D0zUJP69Nji9t1P2lknx7kjs2+37/qTp0xPcAGQvAXMI8/y7JvZI8c+V3YjvfleTVk/xokvdI8tcrv+PQx/l+jIxNvmBhuAb8zI83TfLejVTox1sneWySn22g9nuHdm7h/ezMIVn7p3rnzq4tAbM3S/LNSX6nLaw/7p74aUk+O8kjk7x/kr9faIjtstxQ4/duSb4uyfc23eQfmmF4vi9vjNffvPF/XHKjAWYM3re8TJKHJ/mIJD+28nddBmb2Kkm+Ksn9FoL5CyT5osY2PzDJN3bEoQdvjP4HVrbn1OPM349KcrMkn3UEhr33Z8yBGXGS0Hf7CYZQYPZzTRj8/b17cH1veKPGvljg7k0jK2v0YHYKRwFosENhC/b0b0cYlg9tz/eOF2kL8vs6GxzhlRfykVjZZyTByq2fuTC7WBwwe0RzCH/bfVkPZtge53eqRj9/UBIA+2uneukudJ36We8NxozovgKzXz5SyHDe9jnW+1+i08aEDrSdXujvwewUoRJg/ZnWJ972X4/14QPm+dPXEMxePsl3NI10CSutyEj5BklnKDn0YCaR8DlHHrv+8SU10c2B6Llqk7uYGcqKlWljRnyeFu+b/BuYLZ9BbP7hLZmCyQ5ZmScVc/GzDcyW2/YyXPkOLXspI/jbMx2+eXMw95hgZW7Hqn8wya3ac08JZt7/SS0Lf+4y0xSYSf0+OAnD03PeL0lPbX1Ezx6UZxBa/+QyzKZz7mPvSSVNJFSG5Re3aIzlbZtDuX9LwR+r6+fFzDCUJyz4KPYgehP1abe3TvJXTW9SwPnUkWeYn2QSCYax8hZZ5NdoutUrJMGWf72J6RJdar6GTIMDdw8g8nMCuHvVid21lSs8qrGUPxzpk5AR2OjXksVvTX1be867JvHsYaux8++SSbLhp2ww4gFJ7pnkN0754uG7psBMqcD3tIun4nDZFZPxrVqWE3sbAt55fttFfHeJpkJLbUqwfaUk353kVVuJxsccSccqG50azNRHKSMA0rJwu5p5ZoH+Upf5Y8e3a3qNBV/PoEOppPdv5uMvTGQzK4uMIX9AywwKmYATUb5vFXUoTVICoQ7QtXQvdZbGRuX+LZNgUMLBKVmmwjJAPJfUwcoe0rK/uzTp92oJOn0GsjK5p2zFDDmbJWFz3zfjKOxm11dM8k9JOHBz40v2LUkaAzNamfqf+zZPZYDGxL1TL7i1B6j3eoc8+40XLMh6vlDgW5oD+PEkJuKfjrxc+r1qjz651RId0se5e08JZi/cMnlKNN4myc/PdK7GaWhngCSL9ozGYrEmACeqkGzYVZrhvRIr5nkvwltQFtLrNRlADdd/Nlas2r0P//+iJTGUVihctZb0lTTj/32fYta+FQHARufCwbdsUREAdO1YxtA7gao/3rnEnnNzYd+fV6ShdMvcXtrYgiNAnEhVj27MVu0d1rp3ZnYMzGQoDHRlMqdCHPSyxMhTZ1GWGmzXdecBZlUTpF8WylQ2y+CaHNreg3oG45wKzACQCQwAluqslWQaliT4TAseeAGgav0CH9MblW7IDGJlQ0dRYSAdSKg5jDZ6aUXBuBorIFKtxP03bHVkw8xiLXwZwF3hYN8PzwZSY6ynB9fzqigQ8iuzgQVzAF12YgesU4lOXzPXs9Gpb56c3mNgtu8in/JCZ1hTV/oWtpa1XDrgjEHDmWLGaxrrVGBWmTmLfimY0WvVVUmWcKw8eG3Gxlowvb72kV0KAMfArNcjx7J/Nf/H+teDGTAaZn7nnl1gZgeNouipJmTF8DDNpc3uEYz01FsL65uF+kvmdgGZOUcv5rSVp/R1l6IWNYl/ufTjXTcEsz5DCaRsmxjTNHqPsCtc2qcvV/3a3ma7FnLv3deaoH3S4RA7H5L672sW6YEv2bzyXHW+Uxos7LdoHRfKCKt2VZ0vBbMxdlzA/pSmQfUZx1OBWZ+hHAPNGsM+OhKaYTunbvuAWV/uRWfEfDkiOGR8sd0/akC2d93aEMz6BbcrQ0kEVfBIAxI2VI2JweYpCbwWkEwobeEYTTaKpxd+2Ne49raYtfvcg5TFORW+98mXPoP1gq3a2p5Y250IvUu98HmDWT+JZckll7CapVuNLG5b52hZ2t+0DNo3TNhgF5j12/DGMvUFZmPFymuB2VyY2YPUVIayD5dldO0m+d0DJi0JgG3MU/bmcB12MNdqbpmPnMOuVuVe3lO4QTt7n7Yb5KFNq+wZmXUOlzg/yR1j/5ixlwzBrM9QTnmEqmA2YYZhkMG2W4CHkKa1YOc875yxhj/XZ+KurJM6LCnwY+zx27dfc9f3CZMphtMnX4aMF5gRwXkvzMZG47VO1DhmmGm8sAbAY4FYdFgar7ykPKHsajLbOidjZzFo9iQKrX5rYPxdYObSvobSHPKcasIbW8xk2Iah4KFgtjQB0GcopxJMtkTJmiIOUwmCuTnZ/xyY6R8bIwn6MJeccf/S0Lmf2wiOSgisW/kLvc18HyM+wIyWBkskQTh7TujZ2hDMllQT9xkfAioxtd+SUYsWc1u6Z6vQV+ZIenZX02deyd8YoeOIzgJm+2qDU31ams3s64HGForn94x37JrKcq4dUhwTzHqdDIABCGdivcmI5jS3+Iw572xeOXlDM7GH+socmFm4xp9eAwgJ+co4HMFkTkko0OmGzuJQMFtamjG3s6Yv8VmqPc7Z1s+fv53koXZuKtM+fE7NHeO8q2awEgVYn8hEacqfLTyMspI6atkmNeR9wQwrI9rZx9fHvP0HVsZuEkFHrLpP3F2318S6KmDWLxS6kELM2njeswb0fKnXXDKBXXMsMOu3bQkvhZZCY/2X8SsBHbiYi5zZsCkRUuLQF6FyZuQF+0gBxDDzNQdm3uGdnkFspw/Ti0Ua2MiUNHIomFUSSGSxi5XOgVlPOsayvEvHfXhdndYCIJcyf0TCkUtzOxqsUwzYeO2rvZYTMEaT0d4QzOpj3nziha/dyjYYoU+pllFqsFD3fdjSeYDZWQf8rPf1JS9jrKtCH3VTNAQLq2+1kOiUCj3X1CKPAWb6i7ULAyu8rG+yWF+u6SRYPS8t2TF2UMGntkMqhzVbdBQ6GoY1LBadAzNgiI39StNflu4pPBTMjCdnL5u5a31UqDvGuuooJU7NmXdrnndnHsgWk3B2ZVtrXlYJiTk5l0ntkxqYtT9Du4vQRDrE/16e6nW5YSR44xoZglkvKqLZjmj5l3Z1eVnhAXQk+A07UwkE96g9g8Zieo3HI1SOed/rAGZ9Dc1wMNmI+Ok4FZ52jK5XAkFiRqW67TPOZBcKKbB9+oITGKaAeG0w69PsvGqFlzVfgI3+08/IFqq+/dvYMUf+HXsabvuqkEjfh6HHLjDTNzVmkiv6tc9JL2uAWRVOSwKMbU8yRr3c0IdvvV2J4Nj71FHbgEHCxM4FjgM4CKt3JY0wZY6Sc9BkGJ0oK7GgzGV4xFeVkEiWzGVS+4LxsaiOXvexbZuaUL8/kltCxLYuJIkWf7ckNGRhqvq7G7ZRjtWZFfvqD88jvkJE7IJHcLrkmDcr9mFhul4HUVD/D22nPOB1ADP2LvZlMItd0RhVPWuovRMIxlqF78XqAOD7tmr6PzgrXWz3rQ1m5oFCSn3sw8vqZjEPoSPHp5ZsalsTYKqTY4nGVWNmkXu2UNNEL912Lssn2y4ZgRlp9nVW9kymjFMAqmPHlvcOaSwj3YvyU/tuS+/inMb25epTz75Kl7Z2ADBWR/iXOdTfsWa9sot5YR2+TnN4uwpRK5RDSMxNz0Zm7IX9kYlkk6Jl40MKmDugcejgaJSYtVIMmAOwgBiA73+pS7G/u7Qxczw4J2gcjcGN4DwGZiWyov4+EAgxPKFV1mEXsqP8PA4ENUm8FGrPbUC/LmDGW96phVUmi61MAJ9GBuCmspMVvhPOeU0AANws7jUOxFwTzHqdbBhe1sJzjcMJMQa7AcyrKUcH+HhpLJQMgklhc8BLZtf5aO4FAGxLqwVUlfHEKGR/AUBtO8IEHQEvfNnV3FPsByBY5P5oniU6kSwAxIBCxOLZmvGVFeXIh6djENhVCyiinip/kP2jqwF6cwNrt6dTxISdTNmr7G+fNOJBdxUJiKholVOgU6GcnQ82jgNOjMzOgrF3ldOwP7U/LHKXPWv+65c5Z4xEFTCCHccqH4r9kVUwamGoZlyf5ReqjIHZWZ18FdwSdqWKyysPF9u+NU9TJSKHJgDO+p3ncV+F7yaVQTfA2Mha546d4nDG87Db1DttBregHJ3D8wtZ1e+Zw5q/hXrmsoW99oGHxVJkZmmCa5XYVIIOAGPwQJZs4TAIgIZgTIFg1bZh+4BT+AaopprMut+z4F17VervORFKaxO9iBbVkwLAZ/uONcFsuAWDV+PFhmAGjWVzZJOq+TdeiiCLbfTNB4zpQdcJzIYOgPaBOY/pj3vOlWt3uSjDBnNsYu53K2B6NowLB9dyHGVw85f2ifkIyddoNU+EyaQg60qIOPcb1IZb7YT/79ROKhnrF/YHXJxssaSw9pBvG5ZQTe5VXhPMegQFbMS8pZtFr0uYedZBLa8p7KAVSawI5YH/1pZboE7sEK4K2ecYBe3PoqY/zR1vvbwX/38lhkgnpf3sYkFLn13zRPJNGL/091r2zF+Y6d7hiSI9CMsES+YJf5dmgpd+Q39dH+1x4Hb6AM/R3TNrghltw545mo4z5BXNMgwRUUbEZJjSdzYwmx7qoQBKZLYdqDKixF7sdW3mcJbJd9HvKYdLf6HnDstf+v7TrMglIoVjso8KeQnf9oMe0orFDLdAWV/0qgJv/+37FKgDoz5xh7GSdhS5KlKlmyEn9lBik8JKOuVoqHdI50furVIxoaWTTIAZ3YxDF0LbQYBF3gCoa4FZhXy3beluQiVvJl4XuxM8/daYKfHxLGBWqXnC7D41bSvb++iP6wVQEwm4mWw0HpqFTBAhfCqzdfQOXqIX9LtXZNM4BL/7oJws6YNzUJJgTmEnhOZjsg/mI6YrXZDcOORdTsFFIsgyasUk3oTKkkVCQmAGxGiAJCBJENlSSZNK3Clsray772cfOCFctU71dXhKybGmQNW90TeBLGckWQBI1SUCMyH6DQx0LTCrGigTpCqH/RujWYwQdRcl3RfMJBccj4L1eY80NbYie7aUWh9rANZ+brGJ/pfm2nYj9Y9h2Es3Vc6xdl8u+/PMd6K+kgC67ViTpfSrFZ0GPFducNHs4fsAkTnhlGKhIJACAFWFgHjYwSNUxATV2gE2dqnN6q4BIIq3zTOlLOfB/NW9AeXa8VLb0DBmobkxutGJrwVmhw6qAjgMDmUUmm5ts8AxLWCxKlWQkcNmNA7hiU2Qvw7HvysJkklFNK7E914UMDvmxN2evVlgs8CzWqAYDuZZx7NfehttYHbph3D7gM0Ce1tANlzhNW3tEI1u7xcf84YNzI5p3e3ZmwU2C5zMAhuYnczU24s2C2wWOKYFNjA7pnW3Z28W2CxwMgv8Hzpz12VmMIZCAAAAAElFTkSuQmCC\" 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width=\"44\" height=\"20\" style=\"width: 44px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e the smallest Euclidean distance between the trajectory of the projectile and the target \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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\" width=\"298\" height=\"20\" style=\"width: 298px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eis a difference angle, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"55.5\" height=\"17.5\" style=\"width: 55.5px; height: 17.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ean update parameter.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 20.6667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 383.5px 10.3333px; text-align: left; transform-origin: 383.5px 10.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample of algorithm's numerical result:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 80px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 403.5px 40px; transform-origin: 403.5px 40px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003etheta = catapult(25,3,25)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003etheta = \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    0.8431\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 403.5px 10px; transform-origin: 403.5px 10px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e    \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 264.333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 383.5px 132.167px; text-align: left; transform-origin: 383.5px 132.167px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"570\" height=\"259\" style=\"vertical-align: baseline;width: 570px;height: 259px\" 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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function theta = catapult(xd,yd,v0) \r\n  \r\n    global g nu;\r\n    \r\n    g   = -9.81;  % grav. acceleration\r\n    nu  = 0.5;    % air friction coeff.\r\n    k   = 0;      % solver increments\r\n    dt  = 1e-2;   % timesteps\r\n    T   = 10;     % simulation time\r\n    TOL = 1e-2;   % absolute tolerance\r\n    \r\n    [~,y] = ode45(@ODECatapult,0:dt:T,[v0,0,0]); \r\n    \r\n    % solver for optimal angle\r\n    while (e \u003e= TOL) \u0026\u0026 (k \u003e 150)        \r\n        \r\n        %theta = theta + beta;\r\n        \r\n        k = k+1;    % add increment\r\n    end\r\n  \r\n    function dx = ODECatapult(t,x)\r\n        global g nu;\r\n        %% fill in ordinary differential equation %%\r\n    end\r\n    \r\n    function e = EuclideanDistance(y,xd,yd)\r\n        %% fill in computation of smallest euclidean distance %%\r\n    end\r\n    \r\n    function beta = UpdateLaw(y,e,lambda)\r\n        %% fill in update law to update the shooting angle %%\r\n    end\r\nend","test_suite":"xd = 8;\r\nyd = 2;\r\nv0 = 35;\r\ny_correct = 1.446;\r\n\r\nassert(isequal(round(catapult(xd,yd,v0),3),y_correct))\r\n\r\n%%\r\nxd = 15;\r\nyd = 5;\r\nv0 = 35;\r\ny_correct = 1.33;\r\n\r\nassert(isequal(round(catapult(xd,yd,v0),2),y_correct))\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":636373,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2020-10-19T12:41:43.000Z","updated_at":"2025-01-02T11:31:42.000Z","published_at":"2020-10-19T13:39:36.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a capapult that fires a projects into the air with an initial velocity\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_0 \\\\in \\\\mathbb{R}_{\\\\ge 0}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The free-flying projectile is subjected to air friction and a gravitional force. Given a desired target \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$z_d = [x_d, y_d] \\\\in \\\\mathbb{R}^2$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and an initial velocity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ev_0 \\\\in \\\\mathbb{R}_{\\\\ge 0}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, find the optimal shooting angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eof the catapult that minimizes the distance between the target and the trajectory of the fired projectile. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etip 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Consider the states \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the x- and y-position of the projectile, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as the x- and y-velocity. Then, the trajectory of the projectile can be found by solving the following ordinary differential equation (ODE):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x_1} = x_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,     \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_2 = x_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e      \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_3 = -\\\\nu x_3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\dot{x}_4 = -g - \\\\nu x_4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.   \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eg = 9.81\\\\; (\\\\text{m/s}^2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\nu = 0.5\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eis the friction coefficient between the air and the projectile. Use the ode45.m function to compute the trajectory of the projectile with initial conditions \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex(t = 0) = (0,0,v_0 \\\\cos(\\\\theta_k), v_0 \\\\sin(\\\\theta_k))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Plotting \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e vs. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e will result in the x-y trajectory of the projectile, as shown in the figure below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etip 2: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eUse the following update law, to incrementally update the shooting angle \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e        \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_{k+1} = \\\\theta_k + \\\\lambda \\\\, \\\\text{sign}(\\\\theta_{e,k})\\\\,e_k\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ee_k \\\\in \\\\mathbb{R}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the smallest Euclidean distance between the trajectory of the projectile and the target \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez_d\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\theta_{e,k} = \\\\text{atan2}(d_y,d_x) - \\\\text{atan2}(v_0\\\\sin(\\\\theta_k),v_0\\\\cos(\\\\theta_k))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eis a difference angle, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e$\\\\lambda = 0.01$\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003ean update parameter.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample of algorithm's numerical result:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[theta = catapult(25,3,25)\\ntheta = \\n    0.8431\\n    ]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"259\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"570\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" 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this tower of blocks going to fall?\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium02.jpg\u003e\u003e\r\n\r\n*Description*\r\n\r\nGiven a stacking configuration for a series of square blocks, your function should return _true_ if they are at equilibrium and _false_ otherwise. \r\n\r\nThe block configuration for N blocks is provided as a input vector *x* with N elements listing the _x-coordinates_ of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The _y-coordinates_ of each block are determined implicitly by the order of the blocks, which are dropped \"tetris-style\" until they hit the floor or another block. \r\n\r\nAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\r\n \r\nIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\r\n\r\n*Examples*:\r\n\r\nExample (1) \r\n \r\n x = [0 0.4];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01a.jpg\u003e\u003e\r\n\r\nThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return _true_.\r\n\r\nExample (2) \r\n\r\n x = [0 0.6];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01b.jpg\u003e\u003e\r\n\r\nThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return _false_.\r\n\r\nExample (3) \r\n\r\n x = [0 1.5 0.6];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01c.jpg\u003e\u003e\r\n\r\nThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return _true_.\r\n\r\nExample (4) \r\n\r\n x = [0 .9 -.9 zeros(1,5)];\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01d.jpg\u003e\u003e\r\n\r\nThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\r\n\r\nExample (5) \r\n\r\nx = cumsum(fliplr(1./(1:8))/2);\r\n\r\n\u003c\u003chttp://www.alfnie.com/software/equilibrium01e.jpg\u003e\u003e\r\n\r\nThis configuration is stable (see the \u003chttp://en.wikipedia.org/wiki/Block-stacking_problem classic optimal stacking solution\u003e) so your function should return _true_.\r\n\r\n*Display*\r\n\r\nIf you wish, you may display any given block configuration *x* using the code below:\r\n\r\n clf;\r\n y=[];\r\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u003c1)+1]); end\r\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\r\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\r\n 'horizontalalignment','center');\r\n\r\nVisit \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas Block canvas\u003e for a related Cody problem.","description_html":"\u003cp\u003eIs this tower of blocks going to fall?\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium02.jpg\"\u003e\u003cp\u003e\u003cb\u003eDescription\u003c/b\u003e\u003c/p\u003e\u003cp\u003eGiven a stacking configuration for a series of square blocks, your function should return \u003ci\u003etrue\u003c/i\u003e if they are at equilibrium and \u003ci\u003efalse\u003c/i\u003e otherwise.\u003c/p\u003e\u003cp\u003eThe block configuration for N blocks is provided as a input vector \u003cb\u003ex\u003c/b\u003e with N elements listing the \u003ci\u003ex-coordinates\u003c/i\u003e of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The \u003ci\u003ey-coordinates\u003c/i\u003e of each block are determined implicitly by the order of the blocks, which are dropped \"tetris-style\" until they hit the floor or another block.\u003c/p\u003e\u003cp\u003eAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\u003c/p\u003e\u003cp\u003eIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\u003c/p\u003e\u003cp\u003e\u003cb\u003eExamples\u003c/b\u003e:\u003c/p\u003e\u003cp\u003eExample (1)\u003c/p\u003e\u003cpre\u003e x = [0 0.4];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01a.jpg\"\u003e\u003cp\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (2)\u003c/p\u003e\u003cpre\u003e x = [0 0.6];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01b.jpg\"\u003e\u003cp\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return \u003ci\u003efalse\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (3)\u003c/p\u003e\u003cpre\u003e x = [0 1.5 0.6];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01c.jpg\"\u003e\u003cp\u003eThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003eExample (4)\u003c/p\u003e\u003cpre\u003e x = [0 .9 -.9 zeros(1,5)];\u003c/pre\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01d.jpg\"\u003e\u003cp\u003eThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\u003c/p\u003e\u003cp\u003eExample (5)\u003c/p\u003e\u003cp\u003ex = cumsum(fliplr(1./(1:8))/2);\u003c/p\u003e\u003cimg src = \"http://www.alfnie.com/software/equilibrium01e.jpg\"\u003e\u003cp\u003eThis configuration is stable (see the \u003ca href = \"http://en.wikipedia.org/wiki/Block-stacking_problem\"\u003eclassic optimal stacking solution\u003c/a\u003e) so your function should return \u003ci\u003etrue\u003c/i\u003e.\u003c/p\u003e\u003cp\u003e\u003cb\u003eDisplay\u003c/b\u003e\u003c/p\u003e\u003cp\u003eIf you wish, you may display any given block configuration \u003cb\u003ex\u003c/b\u003e using the code below:\u003c/p\u003e\u003cpre\u003e clf;\r\n y=[];\r\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u0026lt;1)+1]); end\r\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\r\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\r\n 'horizontalalignment','center');\u003c/pre\u003e\u003cp\u003eVisit \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas\"\u003eBlock canvas\u003c/a\u003e for a related Cody problem.\u003c/p\u003e","function_template":"function y = equilibrium(x)\r\n  y = true;\r\nend","test_suite":"%%\r\nx = [0 0.6];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [0 0.4];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [0 1.5 0.6];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = cumsum(fliplr(1./(1:16))/2);\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [1.5 2.5 1 0.25 5.5 3.5 -1.5 -0.25 -4 -1.75 6.25 -1.25 0.5 1 -0.5 4.75 -1.25 -1.5 0.5 1.5];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [-1.5 -0.25 -0.25 -1.75 1.75 -2 -5.25 2.25 0.75 -0 0.25 0 -1 -1.5 4.75 -1 4.75 -2.5 3.25 -1];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [1.25 -1.75 -0.75 2.75 -1.5 3.75 1.75 1.5 -0.25 -4 -0.5 -2 1.75 -3.5 -2 -3.75 0 -1.75 1.75 3.25 -1.5 -0.5 1.25 -2 1.5 3 -0.25 1.75 -0.5 2.75 0.5 -4.25 1.5 5.5 3 4.25 2.75 -0.75 0.5 3 3.5 3.25 1.75 1.5 3.25 2.5 5.5 -2 -3.75 -1 5 0.25 -3.75 5.5 1.75 2 1.75 0.5 -3.75 1.5 -0 2 0.5 0 -0.25 0.25 -9 1.75 -3.75 1.25 -3.75 -0 1.75 3.5 -3.75 3.75 -5.75 1.25 3.5 1.5 -2 2 -2.5 -1.5 3 1.75 -1.5 3.25 1.75 -1.5 1.25 -2.5 1.25 -4.25 3.25 -2.5 1.75 1.75 7.25 3.5];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [-0.5 1.5 3.5 0.5 -5.75 2 -1.5 0.25 -0.25 -3 -0.25 3 -3.5 -4.5 1.75 -0.25 0.75 3 0.25 -2.5 2.25 -0.25 1.75 -1.5 -5 -0 -0.5 -2 -0 4.75 -0 2 3.5 1.5 -2.25 3.5 -0.5 4.5 2.5 0.5 1.75 3.5 -0 -2.25 -0.25 -4.75 -2.5 -0.75 -6 2.75 -5 2.25 1 -2.25 -0.75 -0.25 -3.5 0.75 -0 -0.5 -0.5 -1.75 -2 -0.25 -0.25 5 -0.25 -0.75 -0.25 -5 -2 -0.25 -5.5 -5 -0.5 -2 1 -0.75 2 3.25 4.5 2.25 1.25 -0.25 -0.5 -0.25 -2.5 -5 2.25 -2 7.5 6.5 2.25 -0.25 -0.5 7.25 -2.5 1 -2.5 -4.75];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [0.1 0.1 -4.6 -0.4 -1.5 1.6 3 2.7 2.3 -2.7 0.1 -1.7 0 4.4 3.8 -0.4 -2 -0.6 3.3 2.5 -3 -1.7 3.1 2.7 2.7 3.1 -0.4 1.1 -0.2 -0.1 -0.3 2.7];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx = [0.1 0.1 -4.6 -0.4 -1.5 1.6 3 2.7 2.3 -2.7 0.1 -1.7 0 4.4 3.8 -0.4 -2 -0.6 3.3 2.5 -3 -1.7 3.1 2.7 2.7 3.1 -0.4 1.1 -0.2 -0.1 -0.3 2.7 -1.8 2.3];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = [-3.5 2.6 -0.7 -1.1 -2.6 -2 0.8 -0.7 2.7 0.4 -5 3.7 -1.2 -1.3 2.8 0.8 -1.5 -1.8 0 -0 4.8 -1.4 -1.2 -1.5 1 0.2 2.6 1.7 1.6 -1.3 2.1 -1.5 -1.4 2 0.1 -0.1 -0.1 4.6 -3 -0.3 0.2 -1.9 -0 0.1 0 2.1 -1.7 -3.1 -0 0.2 -0.1 -0.5 4.7 -1.8 -0.1 -2.2];\r\nassert(isequal(equilibrium(x),false));\r\n\r\n%%\r\nx = [-2.5 2.6 -0.7 -1.1 -2.6 -2 0.8 -0.7 2.7 0.4 -5 3.7 -1.2 -1.3 2.8 0.8 -1.5 -1.8 0 -0 4.8 -1.4 -1.2 -1.5 1 0.2 2.6 1.7 1.6 -1.3 2.1 -1.5 -1.4 2 0.1 -0.1 -0.1 4.6 -3 -0.3 0.2 -1.9 -0 0.1 0 2.1 -1.7 -3.1 -0 0.2 -0.1 -0.5 4.7 -1.8 -0.1 -2.2];\r\nassert(isequal(equilibrium(x),true));\r\n\r\n%%\r\nx =[0 .9 -.9 zeros(1,8)];\r\nassert(isequal(equilibrium(x),true))\r\n\r\n%%\r\nx =[0 .9 -.9 zeros(1,6)];\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = repmat([0 .7 -.7 0],1,2);\r\nassert(isequal(equilibrium(x),false))\r\n\r\n%%\r\nx = repmat([0 .6 -.6 0],1,2);\r\nassert(isequal(equilibrium(x),true))\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":43,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":3,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-10-02T08:49:59.000Z","updated_at":"2013-10-08T00:22:34.000Z","published_at":"2013-10-02T09:44:07.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/media/image2.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId3\",\"target\":\"/media/image3.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId4\",\"target\":\"/media/image4.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId5\",\"target\":\"/media/image5.JPEG\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId6\",\"target\":\"/media/image6.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIs this tower of blocks going to fall?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDescription\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a stacking configuration for a series of square blocks, your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e if they are at equilibrium and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efalse\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e otherwise.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe block configuration for N blocks is provided as a input vector\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with N elements listing the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex-coordinates\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of the left-side of each block. The blocks are square with side equal to 1 (so the i-th block left side is at x(i) and its right side is at x(i)+1). The\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ey-coordinates\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e of each block are determined implicitly by the order of the blocks, which are dropped \\\"tetris-style\\\" until they hit the floor or another block.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAll blocks are identical (same dimensions and mass) and perfectly smooth (friction is to be disregarded).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIntermediate positions may be unstable. You are only required to determine whether the final configuration is stable.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 0.4];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.4 1). This configuration is stable so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (2)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 0.6];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe first block bottom-left corner is at (0,0) and the second block falls on top of it, with its bottom-left corner at (0.6 1). This configuration is unstable (the second block will fall) so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003efalse\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 1.5 0.6];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId4\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe three block bottom-left corner coordinates are (0,0) (1.5,0) and (0.6,1). This configuration is stable so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ x = [0 .9 -.9 zeros(1,5)];]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId5\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis configuration is unstable, but note that if instead of five we add a few more blocks on top of this at the 0 position that will keep the tower from falling!\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eExample (5)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = cumsum(fliplr(1./(1:8))/2);\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId6\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis configuration is stable (see the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Block-stacking_problem\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eclassic optimal stacking solution\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e) so your function should return\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etrue\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eDisplay\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf you wish, you may display any given block configuration\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e using the code below:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ clf;\\n y=[];\\n for n=1:numel(x), y(n)=max([0 y(abs(x(1:n-1)-x(n))\u003c1)+1]); end\\n h=arrayfun(@(x,y)patch(x+[0,1,1,0],y+[0,0,1,1],rand(1,3)),x,y);\\n text(x+.5,y+.5,arrayfun(@num2str,1:numel(x),'uni',0),...\\n 'horizontalalignment','center');]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eVisit\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1910-block-canvas\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eBlock canvas\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for a related Cody problem.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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Graph: Wichmann Rulers","description":"This Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003chttp://www.azspcs.net/Contest/GracefulGraphs Graceful Graph Contest\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\r\n\r\nAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003chttp://oeis.org/A193802 Optimal Wichmann Ruler\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\r\n\r\nThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\r\n\r\nFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\r\n\r\n*Input:* P  (Number of Points on the ruler)\r\n\r\n*Output:* S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\r\n\r\n*Notes:*\r\n\r\n  1) A W(r,s) does not guarantee all deltas can be generated\r\n  2) For any P there are multiple W(r,s) solutions \r\n  3) P=5 solution is 9, readily solved by brute force\r\n  4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n  5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun ","description_html":"\u003cp\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u003e13.  This Challenge is related to the \u003ca href = \"http://www.azspcs.net/Contest/GracefulGraphs\"\u003eGraceful Graph Contest\u003c/a\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u003e13.\u003c/p\u003e\u003cp\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points.\r\nAn \u003ca href = \"http://oeis.org/A193802\"\u003eOptimal Wichmann Ruler\u003c/a\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/p\u003e\u003cp\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u003e=0 and integer).\u003c/p\u003e\u003cp\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/p\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e P  (Number of Points on the ruler)\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/p\u003e\u003cp\u003e\u003cb\u003eNotes:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003e1) A W(r,s) does not guarantee all deltas can be generated\r\n2) For any P there are multiple W(r,s) solutions \r\n3) P=5 solution is 9, readily solved by brute force\r\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\r\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun \r\n\u003c/pre\u003e","function_template":"function s=Graceful_Wichmann(n)\r\n  s=0;\r\nend","test_suite":"%%\r\ntic\r\nn=17;\r\nexp=101;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=19;\r\nexp=123;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=23;\r\nexp=183;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=29;\r\nexp=289;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=31;\r\nexp=327;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=37;\r\nexp=465;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=41;\r\nexp=573;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=43;\r\nexp=627;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=47;\r\nexp=751;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=53;\r\nexp=953;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=59;\r\nexp=1179;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=61;\r\nexp=1257;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=67;\r\nexp=1515;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=71;\r\nexp=1703;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=73;\r\nexp=1797;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=79;\r\nexp=2103;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=83;\r\nexp=2323;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=89;\r\nexp=2669;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc\r\n%%\r\nn=97;\r\nexp=3165;\r\nS=Graceful_Wichmann(n);\r\nassert(S(end)==exp)\r\ndelta=abs(repmat(S,n,1)-repmat(S',1,n));\r\nassert(length(unique(delta(:)))==S(end)+1)  % zero increases delta unique\r\ntoc","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-09-23T01:30:25.000Z","updated_at":"2013-09-23T13:04:40.000Z","published_at":"2013-09-23T04:00:18.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to find maximum size Graceful Graphs via Wichmann Rulers for P\u0026gt;13. This Challenge is related to the\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.azspcs.net/Contest/GracefulGraphs\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eGraceful Graph Contest\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e which Rokicki completed in 97 minutes. The Wichmann Conjecture is that no larger solutions exist for P\u0026gt;13.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn Optimal ruler is defined as having end points at 0 and Max with P-2 integer points between [0,Max] such that the distances 1 thru Max exist by deltas between points. An\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://oeis.org/A193802\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOptimal Wichmann Ruler\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e readily creates solutions that can be tested for number of points and existence of all expected deltas.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe Wichmann difference vector is [Q(1,r), r+1, Q(2r+1,r), Q(4r+3,s), Q(2r+2,r+1), Q(1,r)] where Q(a,b) is b a's, e.g. Q(2,3) is [2 2 2]. The max value is L=4r(r+s+2)+3(s+1) for Points P=4r+s+3, (r and s \u0026gt;=0 and integer).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor W(r,s), W(2,3) creates the difference sequence [1 1 3 5 5 11 11 11 6 6 6 1 1]. The points on the ruler are the cumsum of W with a zero pre-pended to produce S=[0 1 2 5 10 15 26 37 48 54 60 66 67 68], P=14. All deltas from 1 thru 68 can be realized.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e P (Number of Points on the ruler)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e S (Vector of length P of locations on the ruler, 0 thru Max Value and can generate all deltas 1:S(end))\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNotes:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[1) A W(r,s) does not guarantee all deltas can be generated\\n2) For any P there are multiple W(r,s) solutions \\n3) P=5 solution is 9, readily solved by brute force\\n4) P=13 Wichmann is 57 but the best solution is 58. Too big for brute force\\n5) Create Connectivity Graph for Cases, like Final Matlab Competition, for Fun]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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