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

1. Aug 24, 2010

PhysicsMark

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
$$F(x)=\frac{k}{x^3}$$
$$\ddot{x}=\frac{d\dot{x}}{dt}=\frac{dx}{dt}\frac{d\dot{x}}{dx}=v\frac{dv}{dx}$$

3. The attempt at a solution

I first perform the following steps:

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

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:

$$x(t)=\sqrt{\frac{kt^2}{mx_{0}^{2}}+x_{0}^{2}}$$

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. Aug 25, 2010

nasu

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).