# Become a Pool Shark with Just a Little Physics

## A brief breakdown of billiards

Pocket billiards (also known as pool) is a game we are all familiar with, where the goal is to hit specific colored balls into the various holes on a flat, cloth-colored table surface by striking a cue ball with a long, wooden stick. Different variations of billiards games exist, the most popular being eight-ball, in which one player attempts to knock all solid balls in while the other player knocks in the striped balls; and nine-ball, in which players knock in balls in numerical order with the number 9 ball being the last.

Hitting balls into a hole with a stick: not really a task that requires too much skill, right?

WRONG!

The underlying secrets to this game are the fundamental principles of physics, and understanding how these principles work and apply to billiards can give you a serious one-up on your next opponent. The key topics of interest are collisions, momentum, energy, forces, and friction, all of which can be taken into consideration to make you one mean, green, pool-shootin' machine.

First, we'll look at the rough equations that are involved in the physics of billiards; then we'll apply them to the various types of shots and skills encountered during a typical game.

## Major Physics Equations Involved

There are two major, fundamental laws we need to look at:

• Law of conservation of energy:
• -This law refers to the fact that in an isolated system, the total energy remains constant. In this case, we are looking at mechanical energy, and the conservation of energy in an elastic collision. This law states that if there are no external forces acting upon the system, the amount of total energy present before the collision is the same after the collision. This is represented by the formula:

½(m1v012) + ½(m2v022) = ½(m1vf12) + (½m2vf22)

• Law of conservation of momentum:
• -This law also describes the properties of an isolated system, and states that in such a system, momentum is conserved. Momentum, p is the product of mass X velocity, with units g*m/s. Whether the collision is elastic or inelastic, total momentum before and after a collision is conserved. This is shown by the expression:

m1v1+m2v2=m1vf1+m2vf2

### Get the Ball Rollin'

The cue ball must be hit with enough force to not only move the ball, but to reach and collide with other balls with sufficient momentum. The coefficient of static friction between the pool table and the cue ball is predicted to be about 0.005 (an assumed coefficient for a polished granite ball rolling across a cloth surface), so the force to get the ball moving must be enough to overcome the normal force (N) exerted by the table and the coefficient of static friction1:

Ff=0.005 x mass x acceleration of gravity
0.005 x 0.17kg x 9.81m/s2=0.008 N

0.008 N is a very small force. The static frictional force is negligible in this case, but kinetic friction is necessary since the surfaces of the pool balls are more or less frictionless, and for the balls to exhibit rotational motion. If there was no friction, the balls would continue moving around indefinitely, which would make for a pretty difficult game of pool.
We can find the coefficient of kinetic friction by looking at how long it takes a ball to slow to a stop across the length of the table, or the negative acceleration. This could be experimentally determined, but I'll estimate that a pool ball rolled at 1.5 m/s comes to a stop at about the length of the table, 2.44m.

ax= (vx2-v0x2)/2δx
ax=-0.461 m/s2
To find fk in terms of ax: Fx= -fk = max Fy=mg
μk=fk/N = -max/mg = -(-0.461m/s2/9.81m/s2) = 0.047