# cbloom rants

## 9/13/2007

### 09-13-07 - 2

Collisions are a really annoying thing in physics. It's one of those cans of worms where if you dig into it you have to keep going and going into insane complexity. You start out with this basic point-mass theory where the collision is instantaneous and you're just doing momentum transfers. Okay, that's nice and simple and elegant, but it's bogus for real objects. Say we want to get into slightly more realism, we want to know something about the force and energy of the collision to measure how "hard" it was. Well, the first thing we can do is just pretend that we know the duration of the collision force being applied, and then we can just measure F = momentum change / duration. Okay, but where did we get that duration? And of course in reality the force isn't constant, it's actually more like a really really tight spring that starts off at zero and gets stronger as the objects push together, and then eases off again as they seperate. Of course the geometry of the objects matters too and how they contact, but we'll keep ignoring that and pretend they're spheres. To get the duration of collision and the way the forces vary with depth we have to model the objects as a bunch of finite cells which are compressible with some type of spring constant (or a gas pressure if you prefer). To actually put that model on a solid first-principles basis we should model the atomic structure of our material and figure out the material properties from that. Yuck!

Anyway, I was thinking about why Rugby tackles are so much safer than American Football (NFL) tackles. For background, in rugby tackles you are required to wrap up and stick to your target; it's a penalty if you just come smashing in and bounce off a guy - the reason it's a penatly is specifically for safety purposes. In the NFL, the really hard hits are generally those types of hits where a defender flies in and bounces off his target. There are several factors at play. I'll try to go through them.

1. Inelastic collisions have lower forces. In rugby tackles you have to wrap and stick so you become a single mass after the tackle, in NFL hits you bounce away like billiard balls. Assume you have two momentums, each P, going straight at each other. In the rugby tackle, the two objects collide and stick together and now have zero velocity. Momentum is conserved, but all the kinetic energy is gone from the system. In terms of the force applies, 2P became 0 in some amount of time T, so we'll say F = 2P/T. Now consider the same situation, but instead the two objects collide totally elastically and bounce away in the opposite directions. In this case kinetic energy is totally conserved, but in terms of forces, 2P became -2P , and the force is F = 4P/T - the collision is twice as "hard".

How big of a factor is this in real games? I believe it's not a huge factor. The thing is, in NFL hits the guys don't bounce away totally elastically; they bounce away with a much lower velocity than they came in with. The hit is "harder" proportional to how fast the players are moving away after a hit, which is generally 10% or less of the incoming speeds. So, this is a small contributor, but not a huge difference.