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Instantaneous Velocity from Displacement and Time values

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KLETECH MOTORSPORTS
KLETECH MOTORSPORTS on 12 Feb 2021
Edited: Bjorn Gustavsson on 12 Feb 2021
I have an array of displacement values:
A = 6.15980273514821 6.14448346902259 6.08845116089753 6.10237740189123 6.11608152736223 6.09595708310386 6.08721312535112 6.08234899002155 6.06714986110611 6.07783516323856 6.07792347114879 6.06942843038110 6.06019133043305 6.06261550608395 6.06392554207175 6.07403400879634 6.07480165775461 6.06023816732260 6.06413215823230 6.05980866775166 6.06188423198093 6.04539247866406 5.68706601480939 4.73891273276360 3.33633000147301 1.53377736474907 -0.520209410928023 -2.51119577960674 -4.41312227437336 -6.03989364014165 -7.19838227617766 -7.84654759197349 -7.92368071295474 -7.47392854770517 -6.51019780560905 -5.07107069256220 -3.36973422578889 -1.50050787991370 0.434155257792691 2.12215009949313 3.53006148407630 4.49610896364585 4.92744726480430 4.84056078059572 4.19929415413135 3.15383591162506 1.71286595583864 0.0390502408958251 -1.66192845643858 -3.33294692803218 -4.80249308136781 -5.90361702482525 -6.60666839598799 -6.83843319677040 -6.59833738833612 -5.93158620238645 -4.88505199941910 -3.58238928898158 -2.06162498335311 -0.427978306324553 0.962092147565693 2.21079406034967 3.14407320606021 3.65032027290402 3.71648804678824 3.36000706616073 2.65928916279476 1.60114590344026 0.338579806471336 -1.05833481215168 -2.38367221913169 -3.62501491627475 -4.59871735106957 -5.27463472098307 -5.58863040574497 -5.53036784838860 -5.12001863794680 -4.40681325596626 -3.46518774940656 -2.33147881794239 -1.14253025773902 -0.00120193018594039 1.05416533403683 1.83655243804809 2.34317552443391 2.51855601770804 2.35120342999738 1.90235109235008 1.20233485658391 0.313322912257294 -0.653990145486640 -1.69393994166169 -2.63711164436859 -3.42076266517382 -4.02060849879423 -4.33535045567851 -4.36763583177268 -4.13384363802889 -3.71579617090779 -3.09613185931849 -2.32897335309416 -1.52312351601432 -0.701703723373771 0.0711680436921752 0.672844202931666 1.10019974546121 1.29362318476486 1.24624021443152 1.00403874870039 0.578535494879244 0.0570974441039399 -0.533999805458487 -1.25937401298772 -1.89751921254205 -2.43726901746249 -2.86612610147998 -3.10413193915387 -3.17648524812527 -3.09292469052149 -2.86041286479119 -2.51737566053157 -2.07827258343221 -1.59714345763504 -1.12131785748468 -0.684853319666407 -0.311635411569174 -0.0471022387485308 0.124943852858797 0.127457738929030 0.0209992133123575 -0.216932487595427 -0.456658041475563 -0.764024038661157 -1.10649998879281 -1.39863622698775 -1.67789062705118 -1.88198237554644 -2.01286200701089 -2.05199182485704 -2.00598298730182 -1.92968767887397 -1.78696046276908 -1.64438292043114 -1.48021633242652 -1.31952886260734 -1.18885615969264 -1.07703421305764 -1.01084132179629 -0.971420999249929 -0.978062595741639 -1.01633916058985 -1.05795798469255 -1.09667325981377 -1.11085346887283 -1.12409055761398 -1.12016261878873 -1.10103924041538 -1.11490152792194 -1.11030180564636 -1.10262398739387 -1.11600110094369 -1.10915545278102 -1.12224635477858 -1.10928290200281 -1.11456953961204 -1.11506577559769 -1.10755461106175 -1.10241423280547 -1.11201076964811 -1.10010826509149 -1.11036709193206 -1.11166169215054 -1.11078017535838 -1.10075679537967 -1.10515068040254 -1.09841841252615 -1.10525068092991 -1.11013377694876 -1.11442014135102 -1.12395438247066 -1.09289999828900 -1.11631092596834 -1.13869108092094 -1.13121778411954 -1.11215444772567 -1.08305854689202 -1.07883522161388 -1.07326282278712 -1.09472438786799 -1.10750902820709 -1.11217002961544 -1.13381869147951 -1.12125917557095 -1.10723556079011 -1.12557827186296 -1.11694476376681 -1.10884208834009 -1.10912360573283 -1.12847109342915 -1.11858837419542 -1.13305857893018 -1.13154828131123 -1.11008600118065 -1.11778945807900 -1.11094000460207 -1.09273674128867 -1.10301981432017 -1.07016391933250 -1.05176885156881 -1.03808129358095
and an array of Time periods for the displacement values
T = 0 0.0667334000667334 0.100100100100100 0.133466800133467 0.166833500166834 0.200200200200200 0.233566900233567 0.266933600266934 0.300300300300300 0.333667000333667 0.367033700367034 0.400400400400400 0.433767100433767 0.467133800467134 0.500500500500501 0.533867200533867 0.567233900567234 0.600600600600601 0.633967300633967 0.667334000667334 0.700700700700701 0.734067400734067 0.767434100767434 0.800800800800801 0.834167500834168 0.867534200867534 0.900900900900901 0.934267600934268 0.967634300967634 1.00100100100100 1.03436770103437 1.06773440106773 1.10110110110110 1.13446780113447 1.16783450116783 1.20120120120120 1.23456790123457 1.26793460126793 1.30130130130130 1.33466800133467 1.36803470136803 1.40140140140140 1.43476810143477 1.46813480146813 1.50150150150150 1.53486820153487 1.56823490156823 1.60160160160160 1.63496830163497 1.66833500166834 1.70170170170170 1.73506840173507 1.76843510176844 1.80180180180180 1.83516850183517 1.86853520186854 1.90190190190190 1.93526860193527 1.96863530196864 2.00200200200200 2.03536870203537 2.06873540206874 2.10210210210210 2.13546880213547 2.16883550216884 2.20220220220220 2.23556890223557 2.26893560226894 2.30230230230230 2.33566900233567 2.36903570236904 2.40240240240240 2.43576910243577 2.46913580246914 2.50250250250250 2.53586920253587 2.56923590256924 2.60260260260260 2.63596930263597 2.66933600266934 2.70270270270270 2.73606940273607 2.76943610276944 2.80280280280280 2.83616950283617 2.86953620286954 2.90290290290290 2.93626960293627 2.96963630296964 3.00300300300300 3.03636970303637 3.06973640306974 3.10310310310310 3.13646980313647 3.16983650316984 3.20320320320320 3.23656990323657 3.26993660326994 3.30330330330330 3.33667000333667 3.37003670337004 3.40340340340340 3.43677010343677 3.47013680347014 3.50350350350350 3.53687020353687 3.57023690357024 3.60360360360360 3.63697030363697 3.67033700367034 3.70370370370370 3.73707040373707 3.77043710377044 3.80380380380380 3.83717050383717 3.87053720387054 3.90390390390390 3.93727060393727 3.97063730397064 4.00400400400400 4.03737070403737 4.07073740407074 4.10410410410410 4.13747080413747 4.17083750417084 4.20420420420420 4.23757090423757 4.27093760427094 4.30430430430430 4.33767100433767 4.37103770437104 4.40440440440441 4.43777110443777 4.47113780447114 4.50450450450451 4.53787120453787 4.57123790457124 4.60460460460461 4.63797130463797 4.67133800467134 4.70470470470471 4.73807140473807 4.77143810477144 4.80480480480481 4.83817150483817 4.87153820487154 4.90490490490491 4.93827160493827 4.97163830497164 5.00500500500501 5.03837170503837 5.07173840507174 5.10510510510511 5.13847180513847 5.17183850517184 5.20520520520521 5.23857190523857 5.27193860527194 5.30530530530531 5.33867200533867 5.37203870537204 5.40540540540541 5.43877210543877 5.47213880547214 5.50550550550551 5.53887220553887 5.57223890557224 5.60560560560561 5.63897230563897 5.67233900567234 5.70570570570571 5.73907240573907 5.77243910577244 5.80580580580581 5.83917250583917 5.87253920587254 5.90590590590591 5.93927260593927 5.97263930597264 6.00600600600601 6.03937270603937 6.07273940607274 6.10610610610611 6.13947280613947 6.17283950617284 6.20620620620621 6.23957290623957 6.27293960627294 6.30630630630631 6.33967300633967 6.37303970637304 6.40640640640641 6.43977310643977 6.47313980647314 6.50650650650651 6.53987320653987 6.57323990657324 6.60660660660661 6.63997330663997 6.67334000667334 6.70670670670671 6.74007340674007 6.77344010677344 6.80680680680681 6.84017350684017 6.87354020687354 6.90690690690691 6.94027360694027 6.97364030697364 7.00700700700701
And i need to get the values of instantaneous velocity at the displacement positions.
I considered using diff and gradient, but what i would get is the average velocity.
The displacement values obtained are experimental, so there's no function I can differentiate to obtain instantaneous velocity.
Is there a way to numerically differentiate the displacement values to get instantaneous velocity, not average velocity?

Answers (1)

Bjorn Gustavsson
Bjorn Gustavsson on 12 Feb 2021
There are busloads of things you could try: interpolate to a denser time-grid (using 'phcip' or 'spline' for interpolation method) and take the gradients from that output, you could try some low-order fliding polynomial fits (quadratic over 3-5 points?) and calculate the gradient from those polynomials. But why would they be that much "better" than gradient?
HTH
  3 Comments
KLETECH MOTORSPORTS
KLETECH MOTORSPORTS on 12 Feb 2021
I was thinking maybe that I could define the time period to be so small that the ensuing value of velocity is near instantaneous?
I'm trying to interpolate values of displacement 0.0001 seconds before and after the actual Time period values, so that when i use , the velocity is very close to instantaneous.
I can change 0.0001 to smaller values if need be.
I think spline interpolation, like you mentioned, would work best?

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