-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: ½m₁v_1i² + ½m₂v_2i² = ½m₁v_1f² + ½m₂v_2f²

Conservation of Momentum: m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

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: ½m₁v_1i² = ½m₁v_1f² + ½m₂v_2f²

Conservation of Momentum: m₁v_1i = m₁v_1f cosø+ m₂v_2f cosØ

0 = m₁v_1f sinø - m₂v_2f 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.

Drawn by Craig Caissie

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. As one last note, since pool tables are not

frictionless these equations are not exact due to a small loss of energy due to friction

but they are close enough that your game will improve if these ideas are taken into

account.

 

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