Built on Facts

A few days ago I saw the movie Rudy for the first time.  It’s a story of a young guy in the 70s who wanted to play for the Notre Dame football team.  He was a small guy, and after tremendous adversity finally managed to get on the practice team to function essentially as a training dummy for the actual team.  His dream was just to sit on the bench as a backup player for just one game.  Not my cup of tea, but good for him.  On the last game of his senor year, he’s not only allowed to play, he’s actually put in at the very end of the game as a defensive player.  He participates in the last two plays of the game - on the last one he sacks the opposing quarterback.

“Ok,” thought I, “I can understand a feel-good story but this is so implausible it defies all logic.  Who writes this stuff?”

Turns out it’s a true story.  Oops!

Turns out there’s also a lot of physics in football.  The most obvious example is projectile motion in the throwing or kicking of the ball.  There’s the added bonus that the aerodynamic shape of the ball makes air resistance less of a factor than in baseball and similar sports with spherical or very lightweight balls.  One of the cleanest particular uses of the football as a projectile occurs during punts and kickoffs.  You might want to pick the angle that gives the maximum possible range so as to best establish field position - but your team might not get to the punt returner before he can start running.  If he’s a good runner this longest-distance strategy might not be the best.  Another alternative is to kick the ball at a higher angle to maximize the time the ball spends in the air.  It won’t go as far but the extra hang time allows your team to get into place to best tackle the returner once he gets the ball.  There’s also the wacky on-ground squib kick, but that’s not really projectile motion.

Taking the y-direction as vertical and the x-direction as downfield, the ball’s motion is governed by

and

Where v₀ is the initial speed of the ball.  We can solve the first equation to find t at y = 0, and plug that into the second equation to find the range.  A little algebra and a trig identity gives a range which I’ll call R.

Now we know that sin has a maximum of 1 where its argument equals 90 degrees.  This means we need 2θ to be 90, which means θ is 45 degrees as expected.  That gives us the maximum range.  but the time of flight T (obtained by solving the y equation for t at y = 0) is

This is a maximum where the angle is 90 degrees.  But 90 degrees is a kick straight up, which common sense and the range equation tells us is not very useful since the ball lands where it was kicked.  The time of flight for a full-range 45 degree kick is only about 5/7 of the same kick angled straight upward.  Maybe some angle in between might fit the bill for a particular situation.  It’s a tradeoff, one considered many, many time on any given Sunday.

So next time you see a football game, don’t be prejudiced and think it’s just something for jocks to bump chests about.  The kickers and quarterbacks are doing real physics without even having to think about it.