# What decides distribution of momenta and energy in rigid collisions?

I recall in Highschool physics receiving a problem of the form "assume 2 spherical bodies" $P_1, P_2$ enter a collision where $P_1$ has momentum $m_1 V_1$ and $P_2$ has momentum $m_2 V_2$, both kinetic energy and momentum are conserved and neither changes any medium except the motion of these bodies.

The solution I though was very simple, we have that

$$m_1 V_1 +m_2 V^2 = m_1 V_1 ' + m_2 V_2 '$$

$$m_1 ||V_1||^2 + m_2 ||V_2||^2 = m_1 ||V_1'||^2 + m_2 ||V_2'||^2$$

But now I realized that if $V_i, V_i' \in \mathbb{R}^n$ where $n > 1$ this system may have multiple solutions of $V_1', V_2'$.

For example if a very large body slams into a very small body both headed directly towards each other, you can either completely transfer momentums and kinetic energy, or partially transfer (some average) of momentum and kinetic energy to each body, (now assuming rigid they cannot have a direction vector towards each other) so they move in opposite directions.

But why is complete transfer so much more common in my intuition and textbooks than say 50% of total energy/momentum entering the little body and the other 50% to the big, or even weirder things like $e^{\pi}$%. From a conservation viewpoint these could all be fair game I would think, yet they don't happen.

What law is deciding this?

• In higher dimensions, you need to know how the bodies are approaching each other, and also know the full expression for the force $F(r)$ between them. Once you know that, you know everything. (Also, I think your examples of different possible collision results aren't actually possible. There's a much easier way to see that multiple results are possible: in 2D and higher, the particles might just miss each other, transferring 0% of their energy.) – knzhou Mar 28 '16 at 5:00
• So that is to the say "two rigid bodies collide at some angle" is meaningless, since alone that doesn't tell you how the forces act. – frogeyedpeas Mar 28 '16 at 5:01
• Well, when we say 'rigid body' we often assume the force is zero when they're not touching and infinite when they do, so there's a well-defined solution. But for two bodies in general, yes. – knzhou Mar 28 '16 at 5:03