Football in Flight: A study of the math and physics of the trajectory of a kicked football

Abstract

Three seconds left, down by two, one player will determine the outcome of the big game; one player will make or break his team’s Super Bowl dream. The kicking game associated with the sport of American football is a crucially important aspect of this American pastime. Many factors affect the outcome of a kick, whether it be a last-second field goal, the game-beginning kickoff, or a fourth down punt. Basic factors affecting the trajectory of the ball include the launch angle and velocity of the kick. A much more complicated factor, often ignored by math and physics classes is the force due to air resistance. Finally, the factor most memorable to the kicker that affects the outcome of a kick is the wind. Crosswind, tailwind, or headwind, this factor is the bane of many kickers’ glory. In this study we explore the launch angle and initial velocity of a kick, the force due to air resistance on a football, and the effect that wind has on the trajectory of a football. Our resulting model approximates the trajectory of a kicked football, whether it be through a field goal, kickoff, or punt. Taking into account air resistance, crosswinds, headwinds, tailwinds, various initial velocities, different kick angles, and any geographic location, our model can plot the trajectory of any kick.