The movement of balls in sports often border on iconic. The sweeping curve ball dropping into a catcher’s mit. The uncanny bend of a football/soccer ball into the corner of the net. The drastic dip of a heavy topspin forehand hitting the court before exploding upward again. In American NFL football, the first thing that comes to mind is the tight, rotating spiral of a pass thrown with pinpoint precision.

A recent paper titled “Modeling the Dynamics of an American Football and the Stability Due to Spin” offers a detailed mathematical model for understanding the motion of an American football, particularly focusing on the dynamics involved in a spiral pass and punt. This model is based on well-established equations used for spinning projectiles, but it is uniquely tailored to account for the specifics of an American football’s motion through the air. 

According to lead author John Dzielski, an engineering professor at Steven Institute of Technology, much of the inspiration for doing the study came from a New York Times article that highlighted a previous study that attempted to explain the flight of a spiral football.

“The Times article cited a study that attempted to explain this behavior, but their analysis was flawed, based on incorrect assumptions. I noticed errors in their examples and realized their conclusions were off,” says Dzielski. “To put it simply, according to Newton’s laws, an object in motion tends to stay in motion. This principle also applies to spinning objects, like a football. When a quarterback throws a spinning football, you’d expect its spin axis to remain constant in space. However, the football’s nose tips over during its flight, which seems puzzling. Despite my background in ballistics, I hadn’t thought much about this phenomenon until I read that article.”

Dzielski builds upon a previous study by William J. Rae and his initial numerical study of an American football’s motion during a spiral pass. Rae’s study considered the spin momentum of the football along with four key aerodynamic effects: drag, lift, overturning moment, and Magnus lift force. Dzielski’s analysis significantly extends Rae’s mathematical model to include all terms necessary to comprehensively describe the dynamics of spinning projectiles in flight. It not only adds to the depth of the study but also addresses the gaps in experimental data, especially concerning aerodynamic effects that result in damping due to angular motion.

In defining the football’s location and orientation, Dzielski establishes a reference frame, crucial for accurately describing the football’s motion. The ‘base’ frame is located at the quarterback’s position when the ball is released, while the ‘ball’ frame is at the football’s geometric center. The orientation of the football is intricately described using Euler angles, focusing on the vector along the ball’s symmetry axis. This precise definition of frames and orientations is key to accurately modeling the football’s dynamics.

Using the aerodynamics of artillery shells as a guide, Dzielski describes the football’s translational and rotational motion, formulating differential equations that incorporate various forces such as gravity, lift, drag, and the Magnus force. These equations account for both the location and velocity of the football, and importantly, they consider the effects of Earth’s rotation on the football’s trajectory. 

“People have been studying artillery for a long time, and in doing so, they’ve identified various aerodynamic effects that influence the motion of an artillery shell,” Dzielski explains. “When encountering a new shape that hasn’t been studied before, the process is quite straightforward. Researchers determine what tests are needed to figure out these unknown coefficients for the new shape. Once these coefficients are known, it’s simply a matter of plugging them into the equations to predict the object’s behavior. This approach shows how a deep understanding of physics can provide insights into the behavior of new shapes, like a football, by applying established principles.”

Dzielski draws parallels between the motion of a football and a gyroscope. A gyroscope set at an angle on a table exhibits a unique precession, rotating about the vertical axis while also oscillating its spin axis away from and towards the vertical. This dual motion, challenging to discern separately by eye, is similar to what a football undergoes. Dzielski describes the phenomenon as ‘wobble’ and is caused by torque. For a gyroscope, it’s the gravitational torque due to its tilt. In a football’s case, it’s aerodynamic torque, especially evident when a football is placed in a wind tunnel at an angle, leading to forces of drag, lift, and a torque that tends to push the nose upwards.

However, unlike the gyroscope, a spiral football’s axis rotates around its velocity vector. 

Dzielski explains the significance, “This vector changes as the quarterback throws the ball; initially pointing upwards, gravity then alters its direction gradually downwards. As a result, the football’s nose rotates around this evolving velocity vector.” 

He found that for a consistent rotation of the football’s nose, there must be a constant torque acting on it, perpendicular to both the spin and its change. 

“This leads to a noticeable drift in the football’s path,” he says. “For a right-handed quarterback, the football’s nose stays slightly to the right of its direction of travel (and to the left for a left-handed quarterback), causing the pass to drift accordingly. This drift can be significant over longer distances. The most conspicuous effects in a football’s flight are this wobble, the alternating orientation of the nose, and the lateral drift, while other influences are comparatively minor.”

Of course, Dzielski’s findings won’t help up and coming quarterbacks throw better spirals or develop the precision they need to rise to the next level. That’s a matter of mechanics and, to a degree, anatomy. However, once the ball is released and takes flight, we now have a better idea of the forces at work that create the signature spiral that makes American footballs so identifiable.

IMAGE CREDIT: Time, NFL.