# How to model perfectly elastic collisions using force?

I want to have elastic collisions in my Newtonian physics simulator, but in the one physics class I've taken we only used momentum when analyzing elastic collisions, and my simulator is totally force-based.

If two pointlike objects are colliding with given velocities and masses, what forces should I have them exert on each other in order to simulate a perfectly elastic collision?

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There's a reason you analyzed such collisions in terms of momentum. The required force is a dirac-delta function of the distance between the particles, which is a strange mathematical object students normally don't encounter until their first quantum mechanics course. –  David H Aug 27 '13 at 4:39
Oh, wow. That's really upsetting. Do you know if there's a good heuristic of some kind? I don't want to break my simulation's "purity" and directly edit velocity. –  Eli Rose Aug 27 '13 at 4:49
Sorry, I'm not the right person to ask for advice on computer simulations. :( –  David H Aug 27 '13 at 5:01
Is this a 1D or 2D/3D simulator? –  Kyle Oman Aug 27 '13 at 18:26
@Kyle It's a 2D simulator –  Eli Rose Aug 27 '13 at 19:24

## 1 Answer

This question seems to be closer to the Computational Science group rather than Physics. In brief, any central potential force would do this - Lennard-Jones, Coulomb (better use screened Coulomb), the exact form doesn't matter; any of these forces will result in motion conserving total mechanical energy and momentum so an elastic "collision" motion will follow inevitably. There is a whole field of computational science called Molecular Dynamics (MD) where details of this are worked out, I suggest checking out http://en.wikipedia.org/wiki/Molecular_dynamics#Potentials_in_MD_simulations

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This comes with some caveats I think. If the time resolution of the simulation is too long, particles can potentially move through each other without an collision-like interaction. This is easy to deal with in 1D (don't let particle order change, go back and decrease timestep instead), but in 2+ dimensions it gets tricky. I'd also recommend an approximated potential that actually goes to zero at some reasonable radius instead of an exponential (or other) decay to save on computation, but whether that's important depends on how many particles are being simulated. –  Kyle Oman Aug 27 '13 at 18:29
The Lennard-Jones potential has a minima point, which might cause attractive forces to develop between particles. Wouldn't really want that when you want to simulate normal force between particles. Although you could cleverly set the minima to be such that the particles seem to touch at that point. –  udiboy1209 Aug 29 '13 at 16:42