I choose my words carefully:
"the head smacking into the inside of the helmet is exactly equivalent to a head smacking off solid concrete as far as dissipating kinetic energy goes"
Key phrase there ;)
It's quite interesting to analyse something like a collision as it gives an insight into some nice bits of physics. If you're head is, with respect to the ground, going and some velocity v in meters per second and has a mass m in kilograms, then the kinetic energy it has is found simply by:
k.e=1/2*m*v^2
which is burned permanently into my brain as 'half mass times velocity squared'. If you want to bring your head and helmet to a stop you obviously need to bring that velocity and hence kinetic energy down to zero. As noted earlier, with very few specific exceptions, you cannot just get rid of energy, it has to go somewhere. We care that it goes into say compressing a helmet rather than right into your skull, but the point is your head has a certain amount of k.e and it has to be gotten rid of one way or another. Having a helmet on doesn't change that fact.
So, you cannot change the k.e needing to be moved in an impact, but what you can change is how it's done. It's a bit simpler to consider momentum, which is very simply mass times velocity, or the derivative of kinetic energy (or k.e is the integral of momentum...hooray maths!). So here's another extremely useful equation that gives a whole bunch of explanations to everyday phenomena:
Force = change of momentum/time of deceleration = (mass*velocity)/time of deceleration
This is good old Newtons second law written in a certain form that works for us. Just as you're moving head has a certain k.e that we need to bring down to zero so too does the momentum need to brought to zero. Now, for k.e we're powerless to do anything about it, but in this equation what we do have control over is the time of deceleration, or what you might call 'impact time'. If you're head smacks of hard concrete, then it stops almost instantly, in other words the bottom of the force equation is very small, which makes the force exerted on your head very big which equals a concussion, basically. But lets say we have some nice soft padding for your head to hit between it and the concrete. This will increase the impact time/time for deceleration and will by the magic of mathematics reduce the force exerted on your poor head. This also explains why suspension forks on a bike softens landing from a jump say, or why jumping onto a bed doesn't hurt as much as jumping onto concrete, or a million other things.
This is what I meant about being careful with terms, a phrase like 'a helmet doesn't reduce the energy of an impact' seems like nonsense, but it's quite true.
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