Relativistic Constant Force Motion Explorer

This Live Script examines 1d and 2d motions of a particle subject to a constant force governed by relativistic mechanics.

This Live Script examines 1d and 2d motions of a particle subject to a constant force governed by relativistic mechanics. The motion of a relativistic particle of invariant mass m subject to a prescribed force is described by the relativistic form of Newton's 2nd law, with the solution . Here, the over-dot denotes differentiation with respect to time and the linear momentum is , where is the velocity and . From the solution , we can find the energy , the velocity and the position vector by integration of the velocity: .

Relativistic constant force motion exhibits features distinct from non-relativististic constant force motion. This script derives the solution for the relativistic case using MATLAB dsolve as an illustration of the use of the symbolic math toolbox and examines the work energy theorem, proper time, and range for one- and two-dimensional motions.

Illustrations include the time for a rocket to uniformly accelerate from rest and then uniformly decelerate to arrive at a destination at rest, and 1d and 2d ballistic motion of a charged particle in a uniform constant electric field. Because acceleration is not in general proportional to the force for two-dimensional motions, the transverse component of velocity approaches zero at large times. An expression for the shape of a ballistic trajectory is provided and illustrated.

This script may interest students and instructors of physics. 'Try this' suggestions are included for further exploration as well as hyperlinks to documentation and references.