How to Solve an Inverse Square Law Problem in Physics?

[antiduh]

Howdy folks, I'm trying to solve a physics problem that I posed to myself one day after a class. My inspiration is a linear particle accelerator.

So, what I want to do is model a particle in one dimension, being attracted by an inverse-square-law force.

|<---------d-------->|
|<--x-->|            |
|       |            |
|-------O------------M
|       ^            ^
        |            |
Projectile           |
                     |
Attracting point source

Object 'M' is the attracting point source, such as a magnet. Object is fixed.
Object 'O' is the object being attracted, obviously free to move.

Distance 'd' is the position of M.
Distance 'x' is the posiition of the projectile.

So, let us assume that the force on 'O' is:

       S
F = -------
    (d-x)^2

Value S is a unitless abstraction of field strength.


The problem is thus: Find a closed-form solution for the position of the object at time 't'.


Any help would be appreciated. I'd prefer someone to just point me in the right direction, and not solve the whole thing for me, but I'll take whatever help I can get.

 

[stunner5000pt]

i have lost touch with how to do this but i think you do this

find the potential first $$V = \int F \cdot dl$$

then find the Lagrangian L = T - V

Use Euler Lagrange equation to solve for this trajectory

but i could be wrong...

 

[stunner5000pt]

an easier way is

$$F = m \frac{d^2 x}{dt^2} = \frac{S}{(d-x)^2}$$

you now have a linear second order DE which you need to solve