# Finding position and velocity from Force (analytic mechanics)

## Homework Statement

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).

## Homework Equations

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

## 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?

nasu
Gold Member
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).