So I went over to the shelf and grabbed my utilitarian Giro bike helmet to see what the foam thickness on it would be. It came out to about 1.8 inches, so let's take it as 2 inches for sake of simplicity.
Now say one morning, I'm out riding my bike at 25mph on my street. Suddenly, out comes this mad old lady from hell, backing out of her driveway in her car without spotting me over her shoulder. Unable to react quickly, (yeah I had a hangover from last night) I can't swerve away in time and CRASH I do right onto the vehicle, with my helmeted head slamming onto glistening sheet metal.
My head just decelerated from 25mph to 0mph in seconds. What was my head's deceleration? Let's assume it is constant deceleration for an ideal condition. The best helmets should offer this condition.
Avoiding lengthy kinematics derivations, remember this equation for constant deceleration :
a = deceleration of the head with a negative sign
vo = initial speed = 25mph ~ 36.7 ft/s
d = distance the head moves after impact before coming to rest, which is the distance the foam crushed = 2 inches = 0.167 ft
Plug in the values and you get a = -4032.6 ft/sec^2 ------------> A
The acceleration due to gravity on the earth's surface is 32.2 ft/sec^2 -----------> B
Dividing A by B, my head just experienced a constant deceleration of 125 times that of gravity (125 g). That is after wearing a helmet!
If you look at the equation for deceleration above closely, the 2d term is in the denominator. Hence, the bigger thickness of foam you have, the lesser the deceleration becomes. If I had 2 times the foam as I originally had, my head's deceleration would be close to halved and I would have more distance to 'take' this deceleration over. But too much foam thickness doesn't yield a good helmet design as it can cause a host of other problems. So there's a trade off. You will also appreciate the fact that since the velocity term is in the numerator, the faster I go, the more g's of deceleration I experience.
So what about the force my head experiences during this impact? Well, here's another simple equation for you to remember, derived from the work and change in kinetic energy relationship :
m = mass of my head and F is negative in sign indicating a retarding force
Say my head weighs about 10 pounds or 4.5 kgs. Now (vo^2/2d) is simply acceleration we found above, which is 125g. Multiply that with 'm' = 10lbs and you get something in the order of 1250 lbs of retarding force smack against my head. To put that into perspective, that's half a metric ton hitting me right where I don't want it. What makes this so uncomfortable for me, even just to realize, is that its doing this to me in a fraction of a second. That fraction of a second yank on my head is a yank on my brain and its internal blood vessels.
The thing to realize is that 125g of force maybe enough to cause brain injury, leave alone anything higher than this. Rarely is constant force ever experienced in a real collision. A more realistic model would perhaps be force following a curve, reaching a peak when the foam is close to being fully crushed. Hence, the ideal helmet assumed here is sort of...well, ideal. As comments have told me, I acknowledge that this is a simplistic model that does not take rotational characteristics of the acceleration into account. Rarely does an impact between the helmet and the road go through the center of gravity, hence causing rotation of the head.
Another thing you will appreciate is that if I were without a helmet, there would be no soft landings at all. If epidemiological evidence suggests that, on averge, chances of serious brain injury are reduced by a factor of 5 by wearing a helmet, I get a big zero by not wearing one. Lost out there, didn't I?
Having said all this, I sincerely believe that in the coming years as we push the frontier into new materials, we will come up with better solutions for this energy management problem in helmets. We will realize the goal for softer landings, constant decelerations and peak forces of lesser magnitude. Until then, we are struck with what's out there in the market. Some of these helmets are improperly designed. On one hand, too stiff foam liners are used that break catastrophically on impact instead of crushing. On the other, to satisfy the goals of lightweight designs, there is also a tendency to select low density foams that absorb much lesser energy than what is required. Hopefully, these issues wil be addressed soon to give sportspeople the protection they need to keep their heads right.
ADDITIONAL READING :
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