Question

A mathematical model for the position x(t) of a moving object is d²x/dt²+sin x =0. Use a numerical solver to graphically investigate the solutions of the equation subject to x(0)=0, x^'(0)= x₁, x₁≥ 0. Discuss the motion of the object for t≥ 0 and for various choices of x₁. Investigate the equation

d²x/dt²+ dx/dt+sin x =0

in the same manner. Give a possible physical interpretation of the dx /dt term.

Short answer

The solution of the above question is isthat you can use WolframAlpha’sRunge Method.

Step-by-step solution

Motion in a force field:

We consider the motion of a particle in a random isotropic force field. Assuming that the force field arises from a Poisson field in , d ≥ 4, and the initial velocity of the particle is sufficiently large, we describe the asymptotic behaviour of the particle

Solving using graphs:

Graphs of motions are provided:

d²x/dt²+sin x =0

The motion of the given equation appears to be periodic:

The graphs for x₁=x’(0)=0.5,1.0,1.5 are shown in the given figure;

d²x/dt²+ dx/dt+sin x =0

The graphs for x₁=x’(0)=0.5,1.0,1.5 are shown in the given figure;

Hence, the final answer isthat you can use WolframAlpha’sRunge Method.