Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Newton's Second Law Particle Problem

  1. Feb 9, 2010 #1
    1. The problem statement, all variables and given/known data

    4. A particle of mass m is experiences a Force that attracts the particle following
    F(x) = -k*x^-2
    (an inverse square relationship) where x is the distance from an origin at x = 0 and k is a
    positive constant. If the particle is released from rest at x = d, how long does it take to reach the origin?

    2. Relevant equations

    F = ma = m(dv/dt)

    3. The attempt at a solution

    Equating newton's second and the Force equations yields -k*x^-2 = m(dv/dx)(v).

    After seperating the variables and integrating once, I got V = sqrt(2k/m)*sqrt((1/x)-(1/d)).

    My boundary condition was that at x = d, v =0 for the integration. Setting V = dx/dt and trying to solve for x(t) is unfathomable, so I'm not sure how to continue here, or if I have been taking the wrong approach. I've double checked my math as well, so I don't believe the error is there, but I could always be wrong.

  2. jcsd
  3. Feb 10, 2010 #2


    User Avatar
    Homework Helper

    Let the velocity of the particle be v when it passes the origin.
    The kinematic equation becomes
    v = vo + at. But a = -k/m*1/x^2 and vo = 0
    dx/dt = -k/m*1/x^2 or
    x^2*dx = -k/m*t*dt
    Find integration between the limits x = d to x = 0 and find t.
  4. Feb 10, 2010 #3


    User Avatar
    Science Advisor
    Homework Helper

    Hi skyrolla! :smile:

    (have a square-root: √ and an integral: ∫ and try using the X2 tag just above the Reply box :wink:)

    If you need to solve ∫ √x/(√x - a) dx, just make the obvious substitution. :wink:
  5. Feb 10, 2010 #4
    I thought that kinematic equation was only applicable to a constant force, not a position-dependent varying one?
  6. Feb 10, 2010 #5


    User Avatar
    Homework Helper

    In the kinematic equation we have taken into account the position dependence of the acceleration. And we have taken the integration to find t. So it must work.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook