Football Physics

American football is not only a game of strategy and physical prowess but also a fascinating demonstration of the principles of physics in action. From the spiraling motion of a thrown football to the powerful collisions between players, every aspect of the game is governed by the laws of physics.

What makes football unusually rich for a physicist is that it puts several different problems on the same field at the same time: rigid-body collisions in the trenches, gyroscopic motion in the air, projectile motion on every kick, and the ideal gas law inside the ball itself. This page works through each of them, with the real numbers.

Throwing a Football

Collisions

Momentum is a vector describing a "quantity of motion". A football player with more momentum is hard to stop, which is desirable. As momentum is the product of mass and velocity, a player can get more momentum by either increasing his mass or by running faster. The training program of many football players is geared towards this.

The interesting consequence is that speed is often the cheaper route. A 110 kg lineman moving at 5 m/s carries 550 kg·m/s of momentum. A 90 kg running back moving at 8 m/s carries 720 kg·m/s — nearly a third more, despite giving away 20 kg. Adding mass costs a player acceleration and agility; adding velocity costs nothing else.

In a head-on tackle both players feel the same size force in opposite directions, as Newton's third law requires. What differs is the effect of that force, because the lighter player accelerates more for the same force. Total momentum is conserved, so if the tackler and ball carrier lock up and move off together, their common velocity is set by the vector sum of what each brought into the collision.

This is also why coaches obsess over pad level. A player with a low centre of mass has a shorter moment arm between the point of contact and his centre of gravity, so an opponent's force generates less torque about him. He is harder to rotate, and rotating a player is how you put him on the ground.

The energy in a hit

A running player has kinetic energy, which is equal to ½ mass × velocity squared. Each player in a tackle has their own kinetic energy, and because velocity is squared, speed matters far more here than in the momentum calculation.

Impulse: why the duration of a hit matters more than the energy

Energy tells you how big the event is; impulse tells you how much it hurts. Force multiplied by the time it acts equals the change in momentum, so for a given change in momentum the average force is inversely proportional to how long the collision lasts.

Bringing our 720 kg·m/s running back to a stop over 0.15 seconds — a wrap-up tackle where both players give — needs an average force of about 4,800 N (around 1,080 lbf). Stopping the same momentum in 0.015 seconds, the kind of abrupt contact you get from a rigid, square-on collision, needs about 48,000 N (around 10,800 lbf). Nothing about the player changed. Only the time did.

Every piece of protective equipment in the sport is an attempt to buy milliseconds: helmet foam, shoulder pad padding and the soft-shell covers now common in practice all work by extending Δt so the peak force falls. It is worth being careful here, though — reducing peak linear force is not the same as preventing concussion, which also involves rotational acceleration of the head, and studies of soft-shell helmet covers have produced mixed results on concussion rates even where they clearly soften the impact.

Throwing a Football

The shape of a football (a "prolate spheroid") gives it unique physical characteristics. Obviously if the ball is thrown sideways it will not go very far, as there will be tremendous drag. However, when thrown with the point of the ball forward it cuts through the air, the drag is reduced, and it travels great distances. In addition, the spin that a quarterback puts on the ball makes it a gyroscope and increases stability. A good quarterback can put a spin of 600 rev/min on the football. See the discussion about the longest throws in football.

Some numbers to anchor that. A regulation NFL ball has a mass of about 0.41 kg and is around 11 inches long, with a semi-major axis of roughly 0.14 m and a semi-minor axis of roughly 0.086 m. Its moment of inertia about the long axis is about 0.00198 kg·m², appreciably smaller than the 0.00283 kg·m² about a transverse axis — which is precisely why spinning it about the long axis is the stable choice.

At 600 rev/min the ball turns 10 times a second, giving it an angular momentum of roughly 0.12 kg·m²/s. On a 50-metre pass thrown at 20 m/s the ball completes about 25 full revolutions before it lands. That stored angular momentum is what resists the aerodynamic torque trying to tumble the ball end over end.

An average NFL pass leaves the hand at about 20 m/s (45 mph); the hardest throwers reach 25–27 m/s (56–60 mph). Drag at these speeds is modest compared with the ball's momentum, which is why a well-thrown deep ball holds its shape in the air so convincingly.

The tight-spiral paradox

Here is a puzzle that took physicists two decades to settle properly. Watch a long pass: the ball leaves the hand nose-up, and by the time it arrives the nose is pointing down, tracking the arc of the trajectory as though it were riding on a rail.

That should not happen. The oncoming air strikes the underside of the ball, which ought to torque the nose upward and start a backward tumble. A non-spinning ball does exactly that. So why does the spinning ball obediently follow its own flight path?

Timothy Gay, together with William Moss and Richard Price, published the resolution: gyroscopic precession. A spinning body does not move in the direction a torque pushes it — it moves at right angles to that push. The aerodynamic torque, combined with the ball's angular momentum and the steadily changing direction of the airflow as gravity bends the trajectory, produces a slow precessional motion that carries the nose downward along the arc instead of flipping it backwards. The precession is real but subtle, running at roughly 340 rev/min against the ball's 600 rev/min spin, and it is slight enough to escape notice even in slow-motion replay.

The practical coaching point falls straight out of the physics: a tighter spiral means more angular momentum, which means the ball is better able to hold a small angle of attack against the airflow, which means less drag and a longer, flatter, more predictable ball. A wobbling pass is not just ugly — it is aerodynamically expensive.

An aside: the ball as a pressure vessel

The ball is inflated to 12.5–13.5 psi above atmospheric pressure, and the ideal gas law does not care about the rule book. Take a ball inflated indoors at 70°F and carry it out to a 45°F field. Working in absolute pressure and absolute temperature, the pressure falls in proportion to the temperature ratio:

That is a drop of roughly 1.3 psi from cooling alone, with no interference of any kind. It is a nice reminder that a fair amount of apparent sporting mystery is simply physics doing its job.

Kicking and Punting

A kick is the cleanest projectile-motion problem in the sport, and it comes with a genuine conflict built in. Maximum range from a projectile launched and landing at the same height comes at 45 degrees, and rather less than that once air resistance is included. Maximum hang time, however, comes from kicking as steeply as possible. A punter who maximises distance gives the returner a running start; a punter who maximises hang time gives away field position.

Measured punts show where the compromise lands in practice: launch speeds of about 24–26 m/s at angles of 47–51 degrees, producing hang times of roughly 3.9–4.0 seconds and distances of 38–48 metres.

Air resistance is also far from negligible. Those same punts travelled 24–33% shorter than a drag-free calculation predicts. Interestingly, drag barely touches the hang time — it mostly eats the horizontal distance — which is why simple projectile equations still give decent hang-time estimates while badly over-predicting range.

Place kicking rewards the same nose-first stability as passing, and the records have been moving. In November 2025 Jacksonville's Cam Little converted a 68-yard field goal, breaking a mark that had stood since Justin Tucker's 66-yarder, and he added a 67-yarder later in the same season. Thinner air helps: kicks at altitude carry measurably further, which is why Denver has long been kind to kickers.

Measuring the Physics: Player Tracking

Much of what used to be estimated on this page is now measured directly. Since 2015 every NFL player has worn RFID tracking chips in the shoulder pads, sampling position many times a second, and the ball is chipped too. The result is a continuous record of position, velocity and acceleration for 22 players at once — essentially a kinematics dataset for every play.

The fastest speed recorded in that era is 23.24 mph (10.4 m/s) by Tyreek Hill on a 2016 kick return. At roughly 84 kg, a player moving that fast carries about 870 kg·m/s of momentum — more than our 110 kg lineman, by a comfortable margin, which is the momentum equation making its point rather forcefully.

Key Numbers in Football Physics

Frequently Asked Questions