Football: Mass, Momentum, and Collisions

Tackles (acts of forcing opponents to the ground) and blocks (acts of preventing opponents from interfering with movements toward the goal) are types of collisions in football. The forces that two or more football players exert on one another due to their mass and momentum can be described by the three laws of motion developed by English physicist and mathematician Sir Isaac Newton (1642–1727).

Newton stated within his first law of motion that any object of mass at rest (velocity, designated as v, is zero) will tend to stay at rest, and any object in motion (velocity is not zero) will tend to stay in motion at the same speed and direction (acceleration, designated as a, is zero), unless acted upon by a force.

Applied to football collisions, the more massive a football player (the more he weighs), the less likely he is to be slowed down, sped up, or diverted by an outside force, such as opponents. The first law is often called the law of inertia because the term inertia means the resistance to motion. For instance, the quarterback who desires to score a touchdown when very near to the goal line will often use the quarterback-sneak. In this scenario, the interior offensive line of players is used as a massive wall to block the defensive line so the quarterback can sneak around or over his men who are very resistant to being moved back.

Newton stated within his second law of motion that the acceleration of an object of mass (m) is dependent on the net force (F_net) acting upon the object and the mass of the object itself. In other words, the second law can be stated: F_net = ma. Acceleration refers to how rapidly an object is changing its speed with respect to time. Force is the action that speeds mass up (acceleration) or slows it down (deceleration).

With regard to football, the second law shows how much force is expended when one player blocks or tackles another player. Although weight is usually used when referring to how much a person like a football player weighs, the player's mass is more important when applied to the second law. Mass is a measure of how much matter (total number of atoms) an object possesses. On Earth, one pound of mass is the amount of mass associated with an object that weighs one pound.

Imagine a linebacker who weighs 240 pounds-mass (lbm) hitting a fullback who also weighs 240 lbm and who is running at 9 yd per second. After the collision, the fullback's final speed is zero. From the time the linebacker first touches the fullback to the time his forward motion is stopped is about 0.2 second. The deceleration is −135 ft per second². With the fullback's mass (240 lbm) multiplied by the deceleration (−135 ft/sec²) the resulting net force becomes 32,400 lbm-ft/sec². To produce a result in pounds of force, it is known on Earth that a force of one pound (lbf) will give a one-pound mass (lbm) an acceleration of 32 ft/sec². Therefore, the result ends up as: 32,400 lbm-ft/sec² × 1 lbf/(1 lbm/32 ft/sec²) = −1,013 lbf. In other words, the force is equivalent to about one-half ton in the negative (backward) direction.

Newton stated within his third law of motion that for every action, there is an equal (in size) and opposite (in direction) reaction. Mathematically this can be written as F₁₂ = −F₂₁, where F₁₂ is the force that body 1 exerts on body 2 and F₂₁ is the force that body 2 exerts on body 1. The minus sign shows that the forces are in opposite directions.

Since both players exert the same force on each other during the collision and continue to do so over the same time interval, but in opposite directions, then one player gains exactly the same momentum (mass × velocity) that the other player loses. Therefore, the net change in the momentum of the two players is zero. This is called conservation of momentum. For instance, if one player who is 300 lbm and running at 20 fps collides with a second player who is 190 lbm and standing still (0 fps), then the initial momentum of the two players is due entirely to the running player because momentum of the two players initially is: m₁v₁ + m₂v₂ = 300 lbm × 20 fps + 190 lbm × 0 fps = 6,000 lbm-fps.

After they collide, the initial momentum is conserved so that after the hit their final momentum must also be 6,000 lbm-fps. If, after the collision, the running player rolls on the ground at 4 fps, then the final velocity of the previously stationary player must be: m₁v₁ + m₂v₂ = 300 lbm × 4 fps + 190 lbm × v₂fps = 6,000 lbm-fps.

Solving for v₂, the answer shows that the previously stationary player is now traveling at 25.3 fps.

Even though it does not appear that tackles and blocks are orderly on the football field, each and every one of them is strictly governed by the fundamental laws of classical physics—compliments of Newton's three laws of motion.

see also Football (American); Football injuries.