Football: A Game of Physics

super bowl XLVII

With Super Bowl XLVII mere hours away, final preparations in New Orleans are in full swing. Teams are undergoing their final meetings, the media frenzy is approaching its peak and the referees are checking their rule books one last time. And while coaches are poring over hours of game footage and quarterbacks are practicing rhythm and timing with wide receivers, one key player is largely ignored: the kicker.

Some of the most famous Super Bowl endings – or infamous, depending on your allegiances – have come at the feet of a team’s kicker, but whether the game ends tragically (“Wide Right”) or triumphantly (“Snow Bowl”), the importance of a “clutch” kicker doesn’t become a story unless it has to.

Kickers have quietly played a major role in shaping the course of the 2012-2013 season. In losing to the Baltimore Ravens in week 3 as time expired and overcoming the New York Jets in overtime in week 7, PTC’s hometown favorite New England Patriots have experienced both sides of these game-changing kicks. In fact, more than thirty games this season have been decided by a field goal in the last two minutes of the game or in overtime.

The official NFL game ball, designed by Wilson®, is 10 7/8 to 11 7/16 inches long from tip to tip with a circumference of 27¾ to 28½ inches along the laces and 20¾ to 21¼ inches around the fat side of the ball; at least thirty of these footballs are inflated to an air pressure of 12.5 to 13.5 pounds per square inch before each game. When NFL kickers line up to kick one these footballs, what forces are really at work at the moment of impact that propel the 14 to 15 ounce ball through the air and – ideally – through the goal posts?

Wilson Football metrics

Successfully kicking a field goal requires pinpoint accuracy and sufficient distance and height, but the fickle physics behind the force of impact involves the principles of conservation of angular momentum and conservation of kinetic energy at work during the collision between ball and foot. The principle of conservation of angular momentum dictates that the total momentum of the kicker’s leg and ball before the collision must equal the total momentum of the leg and ball after the collision. Similarly, the total kinetic energy of the ball and kicker before the collision must be equal to the total kinetic energy of the ball and kicker after the collision.

As you can imagine, the convergence of the equations for angular momentum, linear kinetic energy and rotational kinetic energy gets quite messy, and quickly, and while Stephen Gostowski probably isn’t crunching numbers on the sidelines, we can calculate the exact forces at work using PTC’s Mathcad Prime 2.0.

Angular momentum can be expressed by the equation L = Iω where I equals moment of inertia and ω represents angular velocity. The moment of inertia is mass times the length of the axis of rotation that passes through the kicker’s hip joint. Linear kinetic energy is expressed by the formula KElin = ½ mv2 where m is the kicker’s mass and v is his velocity; rotational kinetic energy can be calculated with the equation KErot = ½ Iω2.

Let’s assume that the average NFL kicker weighs about 200 pounds and has 3-foot-long legs (the axis of rotation) that weigh about 35 pounds. As mentioned above, the official Wilson NFL football weighs about 14 ounces. After the ball is snapped, the kicker starts running straight towards it and reaches a velocity of roughly 12.5 feet per second. Plugging these numbers into Mathcad tells us that the football’s initial speed just after being kicked is 165.692 ft per second (112 mph), but it’s still up to the kicker to make sure the ball is going the right way! (The speed should be somewhere between 70-80 mph and the difference could be explained by the less than 70% energy transfer that we assumed.)

Football Kicker metrics

We’ll have to wait and see if the Ravens and 49ers put another kicker into the NFL’s history books during Super Bowl XLVII, but if the game does come down to the wire, the one thing Mathcad can’t help us figure out is the formula for being “clutch.”

Finding Angular Momentum of Football

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