Physics of Pool

When most of us go out to play pool we do not realize how much physics effects our

game. If we took the time to understand at least the basic physics of pool it might be

amazing to what degree we could improve our skills. Most of us already know at least

somewhat the general idea of how to play pool well. Below I will give a

brief description of how physics plays a part in improving you game of pool. So read on

if you care to impress your fellow pool players!

-Basic Momentum & Kinetic Energy

For the purpose of billiards we will not go into great detail as to what momentum

is. Basically though it can be thought of using the following equation;

p = mv

where p = momentum

m = mass of object

v = velocity of object

Kinetic energy is energy associated with the motion of an object. For basic purposes

we can just look at the following equation which relates kinetic energy with mass

and velocity of an object.

K = ½mv2

where K = kinetic energy

When you strike another ball with the cue ball it is almost a perfect elastic collision.

An elastic collision is one in which total kinetic energy as well as total momentum

are conserved within the system. This can be shown by the two basic equations;

Conservation of Kinetic Energy: ½m1v1i2 + ½m2v2i2 = ½m1v1f2 + ½m2v2f2

Conservation of Momentum: m1v1i + m2v2i = m1v1f + m2v2f

where m = mass of object

v = velocity

Since the cue ball has virtually the same mass as the other balls and the velocity of

our second ball will always be zero, since we are striking a static ball with the cue

ball. In addition this is considered a two- dimensional collision. From this we know

that momentum is saved within the y component and within the x component.

Therefore in the case of pool we can rewrite these two equations as:

Conservation of Kinetic Energy: ½m1v1i2 = ½m1v1f2 + ½m2v2f2

Conservation of Momentum: m1v1i = m1v1f cosø+ m2v2f cosØ

0 = m1v1f sinø - m2v2f sinØ

In this last equation the minus sign comes from the fact after the collision ball two

has a y component of velocity in the downward direction from the x-axis. This can

be seen in the following diagram.

The above diagrams show the initial velocity (both x and y directions) of both balls

(Vxi &Vyi) as well as the final velocities (Vxf & Vyf). As we can see Vxi = Vxf (total

of red and blue balls) as well as Vyi = Vyf.