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Velocity as a function of distance problem

  1. Jan 26, 2010 #1
    1. The problem statement, all variables and given/known data

    The velocity of a ball of mass 'm' varies with the distance v(x) = ax-n, where a and n are positive constants.

    Determine F(x), x(t) and F(t).

    2. Relevant equations
    F = ma
    a = dv/dt
    v = dx/dt


    3. The attempt at a solution
    If velocity were the function of time, i would have done it with no problem. I determined the acceleration, which is -nax-n-1.
    For v, i integrated 'a' with respect to t. But velocity's dependence in distance confused me. Any help will be highly appreciated.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 26, 2010 #2
    There is a simple way for these.
    a=dv/dt. This can be written as a=dx/dt*dv/dx.
     
  4. Jan 26, 2010 #3

    rl.bhat

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    Homework Helper

    v = dx/dt = a*x^-n
    So x^n*dx = a*dt.
    Find the integration and then find x in terms of a, n and t.
    Then d^2(x)/dt^2 will give acceleration in terms of a,n and t. from that you can find F(t)
    Acceleration a = dv/dt = dv/dx*dx/dt = dv/dx*v. Then F(x) = ma
     
    Last edited: Jan 26, 2010
  5. Jan 26, 2010 #4
    To rl.bhat

    Thanks. I tried to follow your instruction.
    Integration of x^n*dx = a*dt gave me (1/n+1)x^(n+1) = v. I didn't find x in terms of a, n and t.
     
  6. Jan 26, 2010 #5

    rl.bhat

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    Homework Helper

    (1/n+1)x^(n+1) = v.
    This should be
    (1/n+1)x^(n+1) = at
    x^(n+1) =a(n+1)*t
    x = .........?
     
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