# Football and Physics

I believe we have a Super Bowl coming up. Or, if the NFL is so picky about the use of their trademarks, I believe we have a “Big Game” coming up. As a native south Louisianian, I’m for the eternally long-suffering Saints, who in all their years have never even been to a Super Bowl. That hypothetical situation was really a part of New Orleans culture – instead of “when hell freezes over” it was always “when the Saints win the Super Bowl”. Maybe they’ll finally do it. I’m not holding my breath, but honestly New Orleans is pretty much crazy with joy that they actually made it to the game.

Fig 1. Drew Brees doing a classical dynamics problem.

We can add some physics to the mix, too. Let’s figure out a football’s trajectory in terms of the initial throw. We know that as soon as the ball leaves the player’s hand there’s only one force acting on it – gravity. Well, there’s air resistance too, but a football is pretty aerodynamic so we’ll just ignore it as a first approximation. Gravity acts vertically downward, so the ball will begin accelerating straight down. But there’s no forces acting in the horizontal direction, so according to Newton there’s no change in horizontal velocity. This constant sideways velocity combined with an accelerating downward velocity gives us the arc that we’re all so familiar with.

Some of the energy of the initial throw will give the ball an initial upward velocity, and some will give it an initial horizontal velocity. By some classic trig, the first is the speed of the throw times the sine of the angle of the throw, and the second is the speed times the cosine of the angle. Write these down:

Here x0 and y0 are the initial coordinates of the ball; we’ll let them both equal 0 for convenience. Theta is the angle of the throw, v0 is the initial speed. a is the acceleration due to gravity, which is -g, or -9.8 m/s^2. The minus is because it’s accelerating downward.

Both of these equations contain the time t, which means they describe the x and y positions as a function of time after the throw. If we’re only interested in total range, we can solve one of them for t and plug it into the other. At the end of the football’s arc, it’s y position will be 0. So set y = 0, solve the y equation for t, and plug that into the x-equation. That gives you the x-position at the time at which the football has hit the ground, and we’ll go ahead and call the range R:

The usual thing to do is simplify this a little with a trig identity:

The max the sin term can possibly be is 1, for a throw at a 45 degree angle, as we might have suspected. Now you might notice that the range is proportional to the square of the initial velocity. Throw twice as fast, the ball goes four times as far. Plugging in the numbers, even at a 45 degree angle you’d have to throw at about 30 m/s = 67 miles per hour to throw the 100 yard length of the field. Short passes are much easier, but they’re thrown fast anyway to make up for the fact that a 45 degree angle throw is usually a suicidal move at short range. Lower angles and correspondingly higher speeds are required.

1. #1 Anonymous
January 29, 2010

R ∝ v02, so doubling throw speed quadruples range.

January 29, 2010

One nit-picking correction: Your range formula only works if the start and end points are at the same vertical position. It will work for a case where a pass is caught by a receiver in stride, but doesn’t give you the spot where the pass hits the ground (unless it’s a kickoff, rather than a pass).

3. #3 Abby Normal
January 29, 2010

Regarding the saying down in New Orleans about the Saints winning the Super Bowl, I’m suddenly reminded of the story of Mr. Gorsky. Thousands of people may have to keep promises they never thought they’d have to.

4. #4 Matt Springer
January 29, 2010

Yeesh, sorry. The math and numbers are right, but somehow I wrote that sentence backwards. It’s fixed now.

5. #5 Itasca
January 29, 2010

Too bad the Saints had to get to the Superbowl the way they did. What are the physics of knocking down a QB 19 times without ever sacking him? Roughly like, I would think, the physics of all the O2 molecules moving to one corner of a room so everyone suffocates. Can’t happen randomly.

6. #6 aweb
January 31, 2010

Except no one can throw a football the length of the field, or even that close for the most part, but most Quarterbacks in the NFL can and do probably reach around the 67mph mark fairly routinely. but a football is pretty aerodynamic just isn’t true, and the only thing that makes it travel remotely well through the air is the spin (the “perfect spiral”). Now there’s a more interesting physics problem – factor in the air resistance and the spin. How hard would someone have to throw to reach 100 yards, and what would the ideal angle be given spin and air resistance?

7. #7 CCPhysicist
January 31, 2010

Are you going to wear a paper bag during the game?

The only problem with that formula is that it doesn’t work on earth. Minor detail? Maybe. In my opinion, deriving it (or the more general one that applies when y is not zero) only encourages formula grabbing rather than analysis. Even the smallest non-zero acceleration in the x direction, let alone one that opposes the vector v, messes everything up.

8. #8 Matt Springer
February 1, 2010

“Pretty aerodynamic” is a relative term. Next to whiffle balls and shuttlecocks, footballs are downright frictionless. Next to a bullet fired on the moon, they’re swimming through molasses.

At 67 miles per hour the wind force is not trivial, that’s certainly true. You can imagine driving a car at that speed and holding the ball out the window to feel the force directly. Still, at shorter ranges and lower initial speeds that’s less of an issue and the approximation isn’t so bad.

9. #9 JG
February 2, 2010

Yeah, i also feel that disregarding air resistance in this case is a) cheating b) makes the problem less interesting c) and if we’re doing it, we should at least compute the error and show that it’s accordingly small

10. #10 Joe
February 2, 2010

45 degrees is not the optimal angle, the optimal angle, when accounting for wind resistance is about 37.5 degrees, according to my Calculus prof. Given the cross section of the ball, if it’s not thrown with a spiral the wind resistance factor is really large, especially given that the ball tends to change its cross section (tumble). To keep the equation even sort of simple you want to assume that the area going in to the wind stays constant. It’s a fluid dynamics problem that exceeds my understanding of the physics- I’d like to see it done though.

11. #11 Alabama Fan
February 2, 2010

FOOTBAWWWWW!!!!

12. #12 Joe
February 2, 2010

Does the Kutta-Joukowsky theorem apply here for the force on the ball? A Karman-Trefftz airfoil is more or less a football.

13. #13 Anonymous
February 3, 2010

better late than never with my nit-pick.

While the field can be assumed to be level, it isn’t. Most “natural” football fields are humped slightly in the center to facilitate drainage, only 6″ or so, so when the receiver is running a “down and out” pattern and the QB stays in the pocket, he’s throwing downhill.

Told you it was a nit-pick,

14. #14 karlo pinar
February 11, 2010

I was just a bit curios.. is it possible that in football, the time relative to the moving ball is slower than the time felt by the people..?? could anyone give me examples of time dilation and explain this to me?? Actually, i am a writer bit right now, I really need facts about the theory of relativity. If possible, please visit my site at

15. #15 nofaith
February 23, 2010

applying physics on football is really great..check this site http://www.nfrsbp.webs.com

16. #16 fatima louise
March 4, 2010

i agree.. applying physics on football is such a great thing. :]

17. #17 joshy
October 3, 2010

how does it work again repeat it for me in smaller words

18. #18 BOB
October 29, 2010

u are all gay

19. #19 Anonymous
March 10, 2011

football always comes down because of gravity bet you didnt know that ha ha

20. #20 Anonymous
March 10, 2011

hello world

21. #21 ball valves
March 27, 2011