Science of Football: Newton’s Laws of Motion, Pythagorean Theorem, and More

Get to know the science behind one of world's favorite sports, the game of fast-paced action.
Kashyap Vyas
The photo credit line may appear like thisKen Lund/Flickr

Football is a popular sport not just within the U.S. but all across the globe. Inspired by soccer and rugby, this intense game of action also involves plenty of science.

Right from throwing the ball, the flight of the ball, and running in the field to tackling the ball carriers, the game offers an excellent opportunity to understand some of the important concepts of physics.


Here's an attempt to give you some clarity on what happens on the football field from the perspective of science.

Newton's laws of motion – throwing and catching the ball

Newton's laws of motion are fundamental laws that govern the relationship between the object and the forces acting on it, and the resulting motion of the object due to these forces. Let's find out how these laws apply to a game of football.

Newton's first law of motion

Newton's first law of motion or the law of inertia states that an object at rest remains at rest, and an object that is in motion stays in motion in the direction of movement unless acted by an external force.

When a quarterback picks up the ball and throws it towards wide receivers, the ball will fly along the direction he threw it with a specific speed based on the force he applied to the throw.

According to the first law of motion, if there's no other force on the ball, it will continue to travel in the same direction at the same speed, until the application of an external force.

But we know from our experiences that when we throw objects, they do slow down gradually and come to a halt after a certain point. The case is no different for a football.

Football Practice Session
Source: C Watts/Flickr

But why does this happen? Why does it have to stop?

The reason why football stops moving is because of the first law of motion. While you don't see any external forces acting on the football when the quarterback throws it, there's still a force that's pulling every object with mass to the center of this planet.

This force is known as gravity or gravitational force. But gravity isn't the only force acting on the football.

When you throw a ball, it also experiences air resistance, which is proportional to the square of the ball's speed and its cross-sectional area.

These forces together prevent the ball from continuing its flight at the same speed and cause it to stop and fall down eventually.

Newton's second law of motion

Newton's second law of motion states that force acting on an object is equal to the mass of the object multiplied by its acceleration. Learn more about acceleration here.

Mathematically, this equation is written as;

F = m x a

Conversely, if we know the force applied on an object and its mass, we can find out the acceleration caused by force. Thus, the equation becomes;

a = F/m

Player Throwing Football
Source: David/Flickr

You can use this equation to find out the acceleration experienced by a football thrown by a player in the field.

The equation clearly indicates that the acceleration of a football is inversely proportional to its mass. It means that when you throw a ball with the same force but a heavier mass, the acceleration will be less.

Newton's third law of motion

The third law states that every action has an equal and opposite reaction. You can easily observe this phenomenon in a game of football when a player tries to catch the ball kicked in the air.

When a player catches the ball, it exerts a force on the player, and in return, the player requires exerting a force of equal magnitude but in the opposite direction to bring the ball to rest.

Another excellent example of Newton's third law of motion in action is when a player tries to tackle the opponent and limit the number of yards he can gain. When the collision happens, both players experience equal and opposite force on each other.

Mathematically, the equation becomes;

F12 = - F21

­Where F12 is the force exerted by body 1 on body 2, and F21 is the force exerted by body 2 on body 1. The minus sign indicates that the force is in the opposite direction.

Defender Tackling Ball Carrier
Source: Peteykirch/Wikimedia Commons

Conservation of momentum – blocking and tackling

The third law of motion also introduces the concept of momentum conservation, in which a collision between the two football players demonstrates clearly. Momentum is nothing but a product of an object's mass and its velocity.

The law of conservation of momentum states that, in an isolated system, when two objects with mass collide with specific velocities, the total momentum of the two objects before the collision is equal to the momentum of the two objects after the collision.

In other words, the momentum lost by one object is gained by another object, keeping the total momentum constant in the system. This phenomenon plays a significant role when a player is trying to stop the ball carrier's forward progress.

Mathematically, the momentum equation is stated as,

m1 x v1 = - m2 x v2

Projectile motion – kicking the ball

There's no better sport than football to demonstrate the concept of projectile motion and parabolas.

A projectile is any object thrown or projected into the air, and the only acting force on the object is gravity. Although, in real-life, a projectile flight gets affected by other forces as well, such as wind and drag resulting from air resistance.

When a punter kicks the ball, it becomes a projectile and follows a curved path known as a parabola in mathematics. This curved motion is because gravity is continually decreasing the velocity of the ball from the moment the punter kicks it.

Player Attempting Kick
Source: John Martinez Pavliga/Flickr

When the ball reaches the highest point of its trajectory, the velocity of the football becomes zero, and from there, it begins falling down quickly. Since gravity is constant, the two variables that decide the object's trajectory are velocity and angle from which the object is launched.

A punter's goal is to either kick the ball as far as possible or, at times, increase the hang time, and that is where the science of projectile motion becomes useful.

The ball will travel the farthest when the player kicks the ball at an angle of 45-degree. Similarly, to achieve high-altitude, a punter must try to kick the ball at a 90-degree angle.

Pythagorean Theorem – defender's angle of pursuit

Player Attempting to Block the Runner
Source: KA Sports Photos/Flickr

Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the other two sides of a right triangle. But, how does this theorem relate to football?

The answer lies in calculating the angle of pursuit, which is the distance that the defender must cross to tackle a ball carrier.


A player who can find the best angle of pursuit has the best chance against an opponent. By incorporating a bit of science into the mix, players can reap sizable results from their game.


Football is a high-energy sport watched by people from all corners of the world. However, not many realize the science behind it. From ball dynamics to the angle of kick, everything in football has a scientific side.

Next time when you're watching your favorite team sweating out on the field to take the ball down the field towards their opponent's end zone, pay a little more attention to the dynamics of the game too. You'll be amazed to find the hidden science.

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