It's impossible to talk about football these days without talking about the problem of big hits. The giant dramatic collisions between players moving at high speed are at once one of the great attractions of the sport for (some) fans, and one of the biggest threats to its popularity. The toll these big hits take on the long-term health of players has many fans reconsidering their allegiances, on ethical grounds.
Understanding what's going on with these plays also involves a good deal of physics. Most of this is the simple classical physics of momentum and energy, but the central issues are frequently misunderstood. And thinking about how to properly understand what's going on turns out to lead into thinking about Einstein's theory of general relativity.
The classic example of a problematic hit for the NFL is one where a wide receiver making a catch collides in mid-air with a defender. These tend to end badly for the receiver, particularly when they're focused on making the catch, and can't brace for the impact. The defenders often fare a little better in these (though it's not unusual for a big collision to send both players to the locker room with concussions); as defensive players tend to run a little larger than receivers, when we ask introductory physics students to explain what's happening, they often make the mistake of thinking that the issue is one of force. The larger defender must exert a bigger force on the smaller receiver, leading the the greater chance of injury.
This is a very common mistake, but it is a mistake. In fact, the magnitude of the force experienced by the defender in the course of making the tackle is exactly equal to that experienced
by the receiver being tackled. This was spelled out clearly back in the 1600's by Isaac Newton, whose Third Law of motion says (in Latin):
Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.
(This translates as "To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." (via Wikipedia))
In more modern terms, we understand this as having to do with momentum, which is (for slow-moving objects, anyway) the product of mass and velocity. A force between two objects is an interaction that changes momentum, and that interaction will have the same effect on each object. In the case where we only consider two interacting objects-- two isolated football players in mid-air, say-- this reciprocity between the interactions means that the total momentum of the two will not change-- if the momentum of one increases, the momentum of the other will decrease by a corresponding amount. This conservation of momentum greatly simplifies what is otherwise an extremely complicated problem-- all we need to think about is the momentum right before the two come into contact, and the momentum right after they stop interacting, and not worry about the messy details of the period when the momenta are rapidly changing.
To use a concrete example, say that a 200lb receiver (about 90kg) moving one direction collides with a 250lb (113kg) defender headed in the opposite direction, and the two come to a stop and fall straight down. In this case, conservation of momentum means that the initial momentum of the two moving players must have added up to the same total of the two after the collision-- namely, zero, because they're not moving. In this case, the lighter receiver must've been going slightly faster than the heavier defender-- 5 m/s to 4m/s, say. The change in momentum in each case is exactly the same, meaning that the force experienced by each is the same.
So why is it so natural to think that the smaller of the two players experienced more force? Well, because what we detect as the sensation of force is actually acceleration, or the change in velocity divided the duration of the collision. And in the football collision, the smaller player experiences an acceleration that's about 25% greater-- they slow from 5m/s to zero, while the heavier defender slows from 4m/s to zero, in the same amount of time.
The acceleration is responsible for the sensation of force, as our internal organs slosh around. It's also what's responsible for the concussions that are currently a major concern for NFL players and fans-- concussions come from the brain shifting violently and running up against the inside of the skull, and the greater the acceleration, the greater the damage.
The way to protect against this damage is to reduce the acceleration, which is why football helmets have thick padding on the inside. The padding compresses during an impact, prolonging the duration of the collision and reducing the acceleration. The total change in momentum is the same, but it takes a little longer to happen, and thus reduces the acceleration.
On the flip side, this explains why one of the biggest causes of concussions is collision with the ground. Other players, covered in protective pads, are often more yielding than the playing surface. A collision with the ground produces the same change in momentum as a collision with a player, but it's over faster, leading to a greater acceleration.
In a collision between real players, the acceleration and thus the danger can be much greater than you would expect from the simple math of masses and velocities because humans are not point particles. You can get a "whiplash" effect when a receiver is hit around waist level, as their body bends around the impact, letting their head continue moving as it was for a short time before being suddenly jerked to a stop. The change in the momentum of the player as a whole is just what you get from the simple intro-physics treatment, but the effective duration is longer for points closer to the impact, and shorter for points farther away.
Portrait of Einstein by Alice Reischer, based on a sketch made during a 1939 lecture. Hangs in the... [+]
What's this have to do with relativity? Well, Einstein attributed the origin of General Relativity to the "happiest thought of his life" ("glücklichste Gedanke meines Lebens" in German, which looks way cooler), one day in the Swiss patent office in 1907 or 1908. Like many people working office jobs they consider beneath them, Einstein was daydreaming about a person falling off a tall building, but rather than imagining the defenestration of a specific co-worker, he was thinking about the physics. Specifically, as he described it, he was struck by the realization that "If a person falls freely he will not feel his own weight."
This may not seem like that great a revelation-- after all, it's the basis for many an amusement park ride-- but that's why Einstein was a genius. He recognized that this idle daydream pointed to a deep connection between gravity and acceleration. What we feel as the sensation of weight is a matter of acceleration, just as with any of the forces involved in football collisions. If the feeling of weight can be made to go away by falling freely, then it stands to reason that it could be simulated by an upward acceleration in the absence of gravity. Gravity, Einstein realized, can't be distinguished from any other acceleration, and working out the consequences of that idea led directly to General Relativity.
So, there's a good deal of physics involved in those jarring hits that make NFL fans and players flinch. Not only do they involve frequently misunderstood points about force and momentum, but thinking about why those points are misunderstood makes a surprising connection to one of the greatest theories in the history of physics.
