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.

2. Helmets and pads actually make hits harder. When rugby players collide, it's flesh and cloth against flesh and cloth. Those materials deform and absorb a lot of the energy of the collision, so it's not transfered to the skeletal system. The collision is "soft" in the sense that the objects can sink into each other and the full force doesn't happen all at once, the mysterious "duration of collision" is longer. Helmets and pads are designed to protect, but they're actually very hard rigid objects, so they collide and don't deform and take the full force of the momentum change over a very very short time frame. It's hard to estimate exactly how singificant this duration difference is to the forces being applied, but it seems like a helmet to helmet NFL collision could have a contact duration in the milliseconds, while a flesh on flesh tackle is going to be closer to a full second, which is something like a 100X difference in intensity. They do have padding inside which is designed to deform and not transfer the full load to the body, but the actual force transferred is much much greater. The role of the internal padding is to take the force and transfer it on to your head gradually over time. Designing padding with the ideal characteristics is very difficult and may be part of the problem. This is also why bicycle helmets are designed to be destroyed in collisions - much of the energy of the collision goes into the material destruction of the helmet, and the crumpling spreads out the impact duration over time so there isn't an instantaneous huge force to the head. But helmets and pads are not actually the question - the rugby rule is specifically about not making bouncing tackles, so something else must be at play.

A related factor is the area of contact. We've just been talking about forces, but we could also talk about pressures. The force of contact is spread over some area, and the smaller the area, the higher the pressure. When two helmets hit, they are nearly spherical, and the area of contact is very very small, which means the pressure on that point is immense. The same is true for shoulders bouncing off eachother, or helmet-to-chest, etc. In contrast, in a wrapping tackle, your body is spread out to make the wrap, and while the initial touch maybe just over a point, as you tackle the area of contact grows and the force is spread out over a larger area. Think of two globs of silly putty thrown at each other. The initial contact will be at a point, but that won't carry much force; over time the two putties deform and sink into each other until the area of contact is very large, and that's when they stop moving.

3. Psychology. Obviously aside from the physical factors there are some large behavioral factors. When you're wearing helmet and pads you feel more invincible and will go in harder with more risks. To make a rugby tackle you generally have to lead with your face up into your target which is subconsciously very scary and will keep you from going full force. The other big factor is that in a rugby tackle you have to be worried about technique and penalties and making a good wrap, so you have to go in slower simply to concentrate on form and be in position to make a proper tackle; if you just fly in at full speed you can't control yourself to make a clean hit. This may actually be the biggest factor of all.

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