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Particle Takes Infinite Time to Reach Top of Potential Hill

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data

    A particle is moving in a potential
    [tex]V(R) = \frac{1}{2} \left( \frac{1}{R} - \frac{1}{R^2} \right)^2.[/tex]
    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
    [tex]m\ddot{R} = \frac{-1}{R^3} + \frac{3}{R^4} - \frac{2}{R^5}.[/tex]
    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:
    [tex]2 = \int_0^t \dot{R}(t) dt[/tex]
     
  2. jcsd
  3. Feb 7, 2010 #2

    tiny-tim

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    Hi Proofrific! :smile:

    Standard trick :wink:

    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 :smile:

    (same result either way)

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