The Super Bowl isn't just a football game. It's an opportunity to discuss physics. Let's look at some of the interesting physics concepts that go with the game.
Deflategate and Ball Pressure
I don't know about you, but I'm getting a little weary of the whole "deflategate" thing. In case you missed the controversy, it appears that some of the footballs in the playoff game between the Indianapolis Colts and the New England Patriots had below-acceptable inflation pressure. Now, it's true that if you put a balloon outside on a cold day, the balloon deflates with the colder temperature. Could something like this have happened to the deflategate balls? The answer is: probably not. If you want more details, Chad Orzel has an excellent piece that looks at the physics of pressurized football. He shows experimentally that a ball in a 50°F football game wouldn't drop 2 PSI due solely to the temperature change.
But why does ball pressure even matter? It is claimed that a lower-pressure ball is easier to grip and easier to throw. I'm not a football player, so I don't know for sure. However, you could experimentally look at the effects of pressure on a ball. Let me get you started on this, because you can do this experiment yourself. (It would be a nice science fair project, too.)
The basic idea is to investigate the bounciness of balls as the pressure changes. Here's the plan.
- Get a football, a pump, and a pressure gauge.
- Measure the pressure in the ball and record it.
- Now drop the ball from a known height and record the height it bounces (repeat 5 times to get an average bounce height).
- Repeat ball drop at the same height but with different pressures.
It might help to record a video of the drop to find the bounce height.
If you don't keep the starting height constant, you might want to record the ratio of the bouncing height to starting height. Now you can make a graph of bounce height vs. pressure. This should give you a nice idea of how much the ball pressure matters in a game.
Bonus: Repeat the experiment with a basketball. At least it will bounce more consistently.
Collisions
You can't really have a football game if there isn't a collision. So let's say there is a big guy going towards a smaller (but still big) guy. Who hits the hardest? You might think the bigger guy packs the bigger punch, but that's not quite true.
Let's look at a smaller blue player colliding with a larger red player.
During this collision, the force that the blue player pushes on the red player is the same as the amount that the red pushes on the blue. The only difference is the direction of the two forces, because there is only one interaction between the two players. The two forces must have the same magnitude. This is just the way forces work. It's a lot like distances. The distance from New York to LA is the same as LA to New York (but in the opposite direction).
But clearly something is different in this collision. Everyone knows that the smaller blue player is going to get walloped. The difference is the change in velocity. To see the change in velocity, we must first look at momentum and the momentum principle. Here are two definitions:
The first is the definition of momentum. Yes, it's a vector - that's why it has that arrow over it. I'm not going to talk about vectors, I just don't want the physics geeks to attack me. (Trust me, you do not want to be attacked by angry physics geeks.) Momentum is the product of mass and velocity. That's not so complicated, right? The second line is the momentum principle. This says that the total force on an object is equal to its change in momentum divided by the change in time.
Now for the magic. Remember that the force on blue and the force on red have the same value but in the opposite direction. If I write this in one dimension (so it's not a vector), then I can write two momentum principles.
What happened to the Δt? Well, it was on both sides of the equation and it canceled. However, the point is that change in momentum for the blue player is the opposite of the change in momentum for the red player. Since the red player has a larger mass, he has to have a smaller change in velocity in order to have the same change in momentum as the blue player.
Yes, there is something different when different mass players collide. It's not the force. It's the change in velocity. That's physics. In fact, this collision physics works so well that you can even use it to determine when a player makes a fake flop. Yes, I am looking at you, Jerome Simpson.
Kicking a Football
There's really just one type of play in a football game that you can almost fully model with physics: the field goal kick. Once the ball leaves the kicker's foot, it essentially just has two forces acting on it: the gravitational force pulling it down, and the air resistance force pushing in the opposite direction of the ball's velocity.
If it was just the gravitational force acting on the ball, this would be a fairly simple physics problem. The gravitational force has a constant magnitude that is equal to the mass of the object multiplied by the gravitational field (g = 9.8 N/kg). The gravitational force also has a constant direction: down (for a locally flat Earth). The gravitational force changes the ball's momentum (see the momentum principle). Since both the gravitational force and the ball's momentum depend on the mass of the ball, the ball's mass doesn't matter in regards to it's motion. I know that sort of seems crazy, but it's true.
Without air resistance, the motion of the football would fall under a model we call projectile motion. It would have constant horizontal velocity and a continually changing vertical velocity. But simple also means boring.
What about air resistance? Next time you are in your car, put your hand out the window. You can feel the air pushing against your hand. Hopefully, you will notice the following:
- The faster the car moves, the greater the force the air pushes on your hand.
- If you make your hand have a bigger surface area (like going from a fist to flat hand), the air resistance increases.
- The air resistance also depends on the shape of your hand. OK, you probably wouldn't notice this, but it's true.
Putting this all together, we can use the following model for the magnitude of the air resistance force.
You can probably guess that A is the area of the object, and you would be correct. C is the drag coefficient, a parameter that depends on the shape of the object. And ρ is the density of the air. This is just a model, but it can usually give fairly nice results.
However, once you have a ball with both the gravitational force and the air resistance force on it, the problem is no longer simple. Really, there is only one way to calculate the motion of such a football: a numerical calculation. The whole idea of the numerical calculation is to break the motion into tiny steps of time. During these small time intervals, we can approximate the air resistance force as having both a constant magnitude and direction. This means that it once again becomes something simple. It's simple, but that one time interval isn't that useful. That means that we would need to repeat this calculation many, many times to get the full motion. This is where a computer comes in handy. These tiny problems are so simple that even a computer could do them. (That's true).
Just as an example, here is a plot showing the difference air resistance can make. This is for a kicked football launched with an initial speed of 30 m/s at a 45° angle. Note that you kind of have to guess on the drag coefficient for a kicked football since it could tumble in different ways.
From these two trajectories, you can see that without air resistance the ball would go about 19 meters farther than it does with air resistance.
Since I have modeled football kicks before, let me go over four of my favorite football kick posts.
- What angle should you kick a football to get maximum range? If you remember your introductory physics course you would say that a 45° kick would give the maximum range. This is true only in the case of no air resistance. With air resistance, a 45° kick will cause the ball to slow down too much. By kicking it at a lower angle you can actually get the ball to travel farther. Of course, the optimal angle actually depends on the kicking speed. A ball kicked with a speed of 30 m/s should be aimed just 42° above the horizon.
- Is it easier to kick a field goal in Denver? It seems like it would be easier for two reasons. First, Denver is at a higher elevation which should give a lower gravitational field. Second, Denver is higher making the density of air lower. So, do these things matter? The changes in gravity are insignificant. The change in air density might give a kick an extra 4 or 5 yards of distance. So, it might matter a little bit.
- Does the Corilios force effect a football kick? When an object is moving in an accelerating reference frame, the normal laws of physics don't work unless you add in some fake forces. The most famous of these fake forces is the centrifugal force that you think you feel when you are in a turning car. There is another fake force you need to add when an object is moving towards the center of a rotating reference frame. Guess what? If you are on the Earth, you are on a rotating reference frame. Even more, if you move north or south, you are either getting closer or farther from the axis of rotation. So, if you kick a football north-south there should be a Coriolis force on it. From my calculation, a 50 yard field goal would have a perpendicular deflection of 0.23 inches. So, not much of an effect.
- Is it better to lose 5 yards in order to kick from the center of the field? Finally, a practical question, right? Sometimes you will see a team in field goal range snap the ball and down it in the center of the field in order to get a better angle on the field goal. This moves the ball farther away but gives a better angle. It turns out that if you only lose 1 yard in the play, it will always be better for the kicker. If you lose 5 yards on the play then you only have an advantage if the kick is from 45 feet away from the goal or farther. And yes, this is for NCAA football and not NFL, since the hash marks are farther apart.
If this is not enough physics to get you through the Super Bowl, here are EVEN MORE football physics posts. Have fun.
