During Sunday's Superb Owl, there were some great tweets looking at science with some type of connection to football. Here is one that got me thinking:
It's not completely trivial, but I can get a rough estimate of the deflection of a football. You know, just to check.
Coriolis Force
What is the Coriolis force? Here is the super short explanation of this force.
- The Earth is mostly spherical and rotating.
- Since the surface of the Earth is rotating, it is an accelerating reference frame.
- In order to make the normal physics rules (momentum principle) agree with the view from a rotating frame, you have to add fake forces.
- There are two fake forces (or you can just combine them as one) for rotating frames: the centrifugal force and the Coriolis force.
- The Coriolis force is the result of an object moving closer (or further away) from the axis of rotation.
- Since the Earth is spherical-ish, moving North or South will bring you closer or farther from axis of rotation.
The best example of the Coriolis force can be seen with a smaller rotating platform (instead of the Earth). Here is a great video from MIT that shows this quite well.
If you want to calculate the value of this Coriolis force, use this:
In this form, Ω is angular velocity vector for the rotating frame and v is the velocity vector for the object. Yes, that is the cross product between those two vectors.
Ok. Enough about the Coriolis force. I really just wanted to do a quick calculation.
Numerical Model for Football Deflection
If you want a super quick approximation, you could do the following (which I suspect is what Neil deGrasse Tyson did):
- Use kinematic equations and projectile motion to calculate the initial velocity of a 50 yard football kick. Ignore air resistance.
- Use this velocity along with the latitude of New Jersey to estimate the Coriolis acceleration on the ball.
- Assume this acceleration is constant over the time interval of the kick and use kinematic equations to estimate the horizontal deflection.
- Use unit conversions to convert your answer from meters to inches since most Americans think in inches instead of meters.
If you want a better approximation, you could include air resistance as well as the changing Coriolis force as the ball changes direction. To do this, you really need to use a numerical calculation. That's just what I will do.
The first step is just to get the motion of a football. How fast and at what angle do you need to kick the ball so that it will travel around 50 yards? This isn't such an easy problem - however, I have already solved this one in a previous post. From this, I will use a launch speed of 30 m/s at an angle of 40°.
Ok, let's just jump into this calculation. There is one slightly troubling part - the coordinate system. Typically when you are looking at the Coriolis force, you would put the axis of the Earth's rotation along the z-axis or something. However, I am going to use the following coordinate system:
In my coordinate system, the angular velocity would be written as:
Where of course the θ represents the latitude of the stadium. But wait! Is the MetLife Stadium facing North? No.
If Google Maps is correct, then the stadium is about 11° from North. I wonder why it's not directly North? Surely there is some logical reason. It doesn't matter. I can just include this angle in my numerical calculation by just considering the ball launched with some smaller component of velocity in the Northern direction.
Here is my first output from VPython:
Oh, you want to look at the code too? Here it is. I added a red line so you could see if there was any deflection. There actually IS some deflection but it is too small to see on the display. If I print the final position of the ball (when it hits the ground), I get a final position of 0.00606 meters or 0.239 inches. This a little bit lower than the estimate by Tyson.
For homework, you can try to calculate the Coriolis deviation using my non-numerical calculation method above. I suspect you will get a value that is closer to the 1/2 inch originally reported.
