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Homework Help: Finding position and velocity from Force (analytic mechanics)

  1. Aug 24, 2010 #1
    1. The problem statement, all variables and given/known data
    A body of mass "m" is repelled from the origin by a force F(x). The body is at rest at x_0, a distance from the origin, at t=0. Find v(x) and x(t).


    2. Relevant equations
    [tex]F(x)=\frac{k}{x^3}[/tex]
    [tex]\ddot{x}=\frac{d\dot{x}}{dt}=\frac{dx}{dt}\frac{d\dot{x}}{dx}=v\frac{dv}{dx}[/tex]


    3. The attempt at a solution

    I first perform the following steps:

    [tex]F=ma=m\ddot{x}=\frac{k}{x^3}[/tex]
    [tex]\int{v{dv}}=\int_{x_0}^{x}\frac{k}{mx^3}dx[/tex]
    [tex]\frac{v^2}{2}=\frac{-k}{2mx^2}+\frac{k}{2m{x_0}^2}[/tex]
    [tex]v(x)=\sqrt{\frac{k}{m}(x_{0}^{-2}-x^{-2})}[/tex]

    The next step is to integrate v(x) WRT time. Doing this will solve for the position. My homework states that x(t) should be:

    [tex]x(t)=\sqrt{\frac{kt^2}{mx_{0}^{2}}+x_{0}^{2}}[/tex]

    If I integrate v(x) WRT time, I will get a different answer for position than the equation above. Is my expression for v(x) incorrect?
     
  2. jcsd
  3. Aug 25, 2010 #2
    You should integrate v(t) over time and not v(x).
    As you don't have v(t) you can instead write
    dx/dt=v(x) , separate the variables and integrate both sides.
    I mean, integrate
    dx/v(x)= dt

    You'll get t = some function of x. Then solve for x and you'll get x(t).
     
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