# Particle Takes Infinite Time to Reach Top of Potential Hill

1. Feb 7, 2010

### Proofrific

1. The problem statement, all variables and given/known data

A particle is moving in a potential
$$V(R) = \frac{1}{2} \left( \frac{1}{R} - \frac{1}{R^2} \right)^2.$$
If you plot this, is has a well at R = 1 with height V(1) = 0 and a hump at R = 2 with height V(2) = 1/32. Question: If a particle has energy 1/32, show that it takes log infinite time to escape from the potential barrier.

2. Relevant equations

See above

3. The attempt at a solution

Using either F=-grad(V) or the Euler-Lagrange equation, I get
$$m\ddot{R} = \frac{-1}{R^3} + \frac{3}{R^4} - \frac{2}{R^5}.$$
How do I solve this differential equation?

Ultimately, I want to find an equation for the velocity of the particle so that I can integrate it to find the time to escape. That is, assuming that we're at R = 1 at t = 0, I want to find the time to reach R = 2 by solving:
$$2 = \int_0^t \dot{R}(t) dt$$

2. Feb 7, 2010

### tiny-tim

Hi Proofrific!

Standard trick

either multiply both sides by R' (so eg the LHS is m(R'2)'/2)

or write R' = V, then R'' = dV/dt = dV/dR dR/dt = v dV/dR

(same result either way)

(and btw, this is where 1/2 mv2 comes from in KE)