Integrating equations of motion

[phys2]

Homework Statement

Suppose that the force acting on a particle is factorable into the following forms.

(a) F (x,t) = f(x)g(t)
(b) F (v, t) = f(v)g(t)
(c) F (x, v) = f(x) g(v)

For which of these cases are the equations of motion integrable

Homework Equations

F = md²x / dt²

The Attempt at a Solution

I just need to check whether I am on the right track

What I did was take F = mdv/dt and used the chain rule ( F = dv/dt = dv/dx times dx/dt = v(dv/dx = F).

So since F = v*(dv/dx), and for (a), F is a function of t, it is impossible to separate and integrate the equations of motion?

For (b) it is impossible to integrate the equations of motion because there is again a time variable in F

For (c), it is possible to integrate equations of motion because F is a function of x and v and you can easily separate and integrate equations of motion using F = v*(dv/dx).

 

[anathema]

Your approach is correct. For (a) and (b), the equations of motion are not integrable because the force is a function of time, which cannot be easily separated and integrated. For (c), the equations of motion are integrable because the force is a function of both position and velocity, which can be separated and integrated using F = v*(dv/dx). Good job!