Quote by Acronim
Defennder:
I don't know how I could possibly apply that if the bodies are not circulating around each other, but simply approaching each other like an apple approaches the earth in a straight line due to gravity.

Considering that the bodies have no velocity at t=0. ok, That is a nice question
basically by the simetry of the problem, you can consider a single axe, say x.
Considering that 1 particle is still(to simplify), you have [tex]a_x=\frac{K}{x^2}[/tex] K is a constant. And the problem is there. That is called a differential equation. You have that the second derivative of x(acceleration) is equal to a constant times 1/x^2.
That differential equation has no analytical solution. Basically that means that you can't derivate/integrate/mathtricks to solve it in order to x(t). So, no, you can't just get the equation of movement of that particle. A computer can, however, make a computational solution(an numerical approximation: with the two bodies it can too). If you wanna know more about it, ask here, and i post you a python code with an how to, or just pm me to talk msn.