The game of pool is an excellent showcase of interesting physics. One of the most common occurences in a game of pool is an almost perfectly elastic collision between a moving cue ball and a stationary ball of equal mass. These collisions occur in two dimensions.
One very useful rule of thumb for pool players is the right angle rule: after the collision the cue ball's final velocity and the final velocity of the struck ball will make a right angle (assuming that both are moving).
Note that this rule is only valid in the limit that ball spin can be ignored. If the cue ball has significant spin, then the rule will be violated.
The rule is generically valid for elastic collisions between equal mass objects when one is stationary before the collision. A short proof is to square the magnitude of each side of the vector version of the equation of momentum conservation:
\begin
[ m^
v_
^
= m^
(v_
^
+ 2\vec
_
\cdot \vec
_
+ v_
^
) ]\end
Cancelling the masses and comparing to the equation of kinetic energy conservation will immediately yield the result that
\begin
[ \vec
_
\cdot\vec
_
= 0 ]\end
which implies that one of the objects has zero final velocity or else the objects move at right angles to one another after the collision.
When this rule applies, it is very powerful. The following examples showcase the utility of the right angle rule.
Part A
The cue ball is moving at 4.0 m/s when it impacts the 5-ball, which is at rest prior to the collision. The cue ball exits the (perfectly elastic) collision at an angle of 30 degrees from its original direction of motion. What are the final speeds of each ball?
System: Cue ball and 5-ball as point particles. External forces are assumed negligible compared to the collision forces.
Models: Momentum and Impulse plus Constant Mechanical Energy.
Approach: We begin with a picture. Since the initial direction of motion of the cue ball is not specified, we arbitrarily assign it to move along the x-axis prior to the collision. We also arbitrarily assign it positive x- and y-velocity components after the collision.