Time dependent position given position dependent force

[scoopaloop]

Homework Statement

A particle of mass m is subject to a force F(x) = −kx^−2 (1) that attracts it toward the origin. (a) Determine the potential energy function U(x), defined by F(x) = − d U(x)/dx. (b) Assuming that the particle is released from rest at a position x0, show that the time t required for the particle to reach the origin is t = π sqrt(m/8k)(x_0)^(3/2)

Homework Equations

dt=dx/sqrt(2(E-U)/m)

The Attempt at a Solution

So, I found The potential energy to be k/x using that and the fact that at v=0 at x_0, I get dt=dx/sqrt(2((k/x_0)-k/x)/m). My only problem is integrating this I get a long nasty function that I feel I can't get x isolated. Maybe, I'm being lazy and need to gut through it, but is there an easier way to approach this?

 

[Orodruin]

My only problem is integrating this I get a long nasty function that I feel I can't get x isolated.

If we are to have any chance of helping you, you need to actually show us what you did.

 

[scoopaloop]

I just want to know if this is the correct approach or if there is another way. I'm not asking you to do it for me. I'd imagine you either know how to do this problem or you don't, I don't see how writing some long pain in the butt equation will help you know how to do the problem.

 

[Orodruin]

The approach is fine, but you seem to be doing it wrong. You can easily factor out the dependence on $x_0$ by a change of variables and obtain a dimensionless integral.

You can also argue for the correct dependence of the time on $x_0$, $k$, and $m$ purely on dimensional grounds. There is only one combination of these parameters which results in a time. Of course, you still need to perform the integral to get the correct prefactor.