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How Helmets Work

Nothing could protect Duncan from the emotional whiplash he would soon suffer, but how did his helmet manage to protect him from physical injury? We can answer this question by considering two physical quantities associated with a hit: pressure and impulse. We've talked briefly about impulse before, and we'll return to it in detail in a moment, but let's first consider pressure.

Pressure is caused when a force is applied to a given area. The actual value of the pressure is the force divided by that area: P = F/A. That's why we talk about pressure in units of pounds per square inch (psi). Remember that in chapter 5 we blew up a football to a regulation pressure of 13 psi. Things can get tricky here, though, because usually when we talk about pressure in this context, we really mean pounds per square inch as read by the gauge (psig), as opposed to an absolute pressure (psia). Absolute pressure is the pressure of the ambient atmosphere plus whatever the gauge reads. Atmospheric pressure, in turn, is what we feel as a result of the force of all the molecules in the air hitting our body. This pressure at sea level is roughly 15 psi. As the altitude increases, there are fewer molecules to hit a given area of our skin within a given time. The force per unit area is less, so the pressure decreases. If a football is blown up to 13 pounds, there are 28 (13 + 15) pounds of force pushing outward on every square inch of the inner surface of the ball.

When Kelso slams Duncan's head with his helmet, we can calculate the force of the hit by again using Newton's Second Law. In this case, Duncan's head and helmet, with a mass of roughly 20 pounds, accelerates to a speed of about 25 feet per second. The collision that causes this takes place in something like a tenth of a second. This corresponds to an average force during the hit of about of 160 pounds, but the instantaneous force can be much higher than the average value.

Now think about what would have happened if Kelso had kept his helmet on but Duncan had removed his. The momentum change (impulse) that Duncan's head suffered would be roughly the same with or without a helmet. It's fairly obvious, however, that without the helmet the result would have been catastrophically different. What saved Duncan's head? Part of the answer lies in the fact that hard-shell helmets significantly reduce the pressure the victim's head feels. When it comes to injuries, the absolute force of the hit is not as important as the pressure that the force delivers. The crucial point here is that pressure is the force divided by the area over which it is applied. The force is distributed over the surface of the helmet facing the blow, instead of being concentrated in the area of initial contact between Kelso's helmet and an unprotected skull. The effective area of the collisional contact between the two heads is bigger with helmets because the helmet material is rigid and moves as a single unit to transfer the force, instead of bulging inward only in the region where the force is directly applied.

We can estimate how much the pressure is reduced by considering the ratio of the collisional contact area when Duncan wears his helmet to that when he doesn't. In the first case, the relevant area is about one-sixth the total outside surface area of the helmet. This is the part that actually ends up pushing on Duncan's head to accelerate it -- the surface area under which the pads are actually squeezed so that they exert a force on his head. This area is perhaps one third of a square foot, or 50 square inches. If Duncan isn't wearing a helmet, there is direct contact between his skull and Kelso's helmet, with the area of the contact being closer to 4 or 5 square inches. Thus with helmets, the force of the collision is reduced by something like a factor of 10.

This pressure-reduction principle is the reason nails have pointy ends. If they had two flat ends, it would be very difficult to drive them into a board. A pointed end greatly reduces the area over which the force of the hammer blow is distributed, which provides a corresponding increase in the pressure that the hammer can apply to the board's surface. To illustrate these ideas to my physics classes, I use a fun little torture device: the bed of nails. It consists of a 3/4-inch-thick piece of plywood the size of a small bed, through which are pounded a couple thousand construction nails. The nails are 5 inches long and spaced about an inch apart on the board. The thing looks positively medieval. We put it up on the lecture table and I climb up and proceed to lie down on the thing, after which I continue lecturing. This demonstration makes the point very effectively. If I were to put all my weight on one nail by standing on it in my stocking feet, the pressure of that nail would easily puncture my skin. If I tried to do the same thing by balancing on two nails, one for each foot, the results would be unpleasant as well. As the number of nails is increased, though, my weight is distributed over more and more of these pressure points. By the time we get up to a thousand nails or so, the pressure from each has been reduced enough so that it can't do serious damage to my skin.

The nails have to exert a total upward force equal to my body's weight in order to keep the net force on me equal to zero, so I won't accelerate into the floor. With the bed of nails, though, the force, and hence the pressure, due to each nail is harmlessly small. When I do this demonstration, I often am asked what the "trick" is. There is no trick -- it's just the basic physics principle of distribution of force, and it is what helps football helmets do their job.

Reprinted from Football Physics: The Science of the Game, by Timothy Gay, Ph.D. 2004 Timothy Gay, Ph.D.. Permission granted by Rodale, Inc., Emmaus, PA 18098. Available wherever books are sold, or directly from the publisher at Rodale, Inc. or by calling (800) 848-4735. For more information, please visit Written Voices Radio.