## Monday, October 12, 2015

### Football Forces

Over Fall Break I was watching the NY Giants game with my dad, and the large collisions between the football players of opposing teams, a central part of the game, caused me to think of the physics behind these large, important hits. An example of this was seen when Jon Beason of the Giants attempted to hit Carlos Hyde of the San Fransisco 49ers on Sunday night. Beason, slightly larger than Hyde (by 10 lbs), collided head on with Hyde, resulting in Beason leaving the game early due to a concussion resulting from the helmet-to-helmet hit.

https://www.youtube.com/watch?t=36&v=8BqGypZ6pfI (for actual footage of collision)

A misconception that I first made when I watched this unfold was that the injury was due to force: the larger defender (Beason) must have exerted a larger force on the smaller running back (Hyde). However, after thinking this through, I realized this could not in fact be the reason why Beason was hurt, because Newton’s third law asserts that for every force, there is an equal and opposite force. Thus, the force experienced by the defender while making the tackle is exactly equal to that experienced by the running back being tackled. So, what principle of physics accounts for this injury?

This injury, in actuality, has to do with momentum:

Thus, momentum is equal to the product of mass times velocity. In the scenario of the collision between Hyde and Beason, we are only considering two objects- the two isolated football players in the air. The forces acting on both players are the same (stated above), as well as the time that the force acts upon the 2 players. Additionally, the impulse experienced by both players is the same, since impulse is equal to the product of the sum of the forces and time:

However, impulse is also equal to the change in momentum. Considering that impulse of the two players is the same, it is important to note that the change in momentum is the same. In other words, the total momentum of the two players will not change; if the momentum of one-player increases, the momentum of the other player will decrease by the same amount. Since each player experiences equal and opposite impulses, they also experience equal and opposite momentum changes. Since the total momentum of an isolated system of objects (here, the two players) remains constant, this scenario really comes down to the conservation of momentum between the players, right before they came into contact with each other, and right after they stopped interacting. In short, the change in momentum is 0, so the total momentum before the collision = the total momentum after the collision.

(Conservation of Linear Momentum)

It is important to note that this scenario assumes that this collision is perfectly elastic, in which the players bounce of eachother and momentum and energy are conserved. Thus, we are assuming that the energy lost is small enough that we can ignore it, because the energy lost is never actually 0.

Specifically, Beason weighs 232 lbs (105 kg), and collided with Hyde who weighs 220 lbs (99 kg). The two, traveling in opposite directions, collided, came to a halt, and fell straight down. Applying the conservation of momentum to this scenario, the sum of the initial momentum of the 2 players is equal to the sum of the two momentums after the collision. The final momentum would equal 0, since both players are not moving. The change in momentum for both players is the same (stated above). Additionally, the time of the collision is the same for both players. Then, because  Δp=ΣFΔt, the force experienced by each player is the same. Because the forces are the same, and F=ma, the lighter player (Hyde) experiences a greater acceleration, since his mass is less. Thus, the lighter player (Hyde) must have been going slightly faster than Beason. It therefore makes sense that the greater accelerating player, Hyde, caused the injury on Beason, even though Beason weighs more, since force and momentums are equal.

To protect against the damage of these types of collisions, football players' helmets contain thick padding. The padding condenses during the collision, which lengthens the time of the collision. Because the change in momentum is the same, the net force decreases, and thus reduces the acceleration. In sum, though the total change in momentum is the same, the time is slightly longer, thus reducing the acceleration and protecting against injury. The crumpling of foam in the helmet is thus pivotal for protecting the players' heads. So, maybe if Beason’s helmet had a little more padding, this collision may not have resulted in this game-changing concussion injury.