It may seem like I'm dunking on half of Hubski here including myself...

I'm sure I'm guilty of it, too, but in the spirit of dunking:

Can "headers" in soccer/football cause concussions, based on the maximum force delivered to the players head when it collides with the ball? You can actually solve it with just calculus if you make the right assumptions...

Not actually a physics problem (but even if it were, I'm sure it could only be solved numerically when you actually try to use physiologically meaningful assumptions about the brain, which is very complicatedly anisotropic). It's a medical problem, and can't be modeled theoretically. If it could, then it would be easy to engineer a concussion-proof helmet, which apparently can't be done. A concussion isn't actually a physical phenomenon, it's a medical definition, which basically boils down to: 1. Have you been hit on the head recently? 2. If yes, are you experiencing headache, dizziness, amnesia, etc, etc. If yes, Congrats! you have a concussion.

I have no doubt you can solve the problem if a spherical object of mass M and spring constant K bumps into another rigid spherical object filled with a fluid of mass m and sheer strength V while traveling at speed V, do you reach a threshold compression of C? Unfortunately, that's not what a concussion is, and nobody actually really knows what causes one.

There's definitely some interesting physics there, but as far as i know, no one has figured it out satisfactorily. One of the most popular animal models of traumatic brain injury (it's also in fashion these days to refer to concussion as a mild TBI) is call the "lateral fluid percussion" model. In this model, you open up the animal's skull and use a tube to direct a compression wave of about a couple atmospheres (I think...can't remember the exact number off the top of my head) to the direct top of the brain. The injury it creates is a very focal lesion far from the site of the blast, but always in the same spot (lateral to the blast, hence the name). I suppose that suggests that it creates a travelling wave that bounces around the brain and two or more of the rebounds create a superposition at the site of the lesion, but I'm not really sure about that. I suggested it once in a meeting with a bunch of TBI experts and they just looked at me blankly then kept talking about something else (because they basically don't know anything about physics--I know about enough to be dangerous).

There's something really important there, but no one really know what yet, because the actual cause of injury in a concussion isn't known. The most popular speculations right now are that it either shears some axons ("diffuse axonal injury") or some micro blood vessels (causing very small "petechial hemorrhages"--small enough that they can't be resolved on a CT). Obviously each has a very different cross section and physical properties, so modeling either is different..or maybe both or neither are relevant.

Anyway, nice arguing about something else for a change :D