[Examples from Momentum]
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 (a.) one of the objects has zero final velocity or else (b.) 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 five 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?
Solution
System:
Interactions:
Models:
Momentum and External Force plus [Mechanical Energy and Non-Conservative Work].
Note that the rule we have derived above relies on the fact that the kinetic energy remains constant during the collision. For this reason, we have stated that we are using the mechanical energy model, though it will not be explicitly used in the solution.
Approach:
Diagrammatic Representation
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.
Before Collision |
After Collision |
|---|---|
|
|
Mathematical Representation
Rather than implementing the elastic collision constraint of constant mechanical energy in the usual way, we will use the right angle rule described above. Using that rule plus conservation of y-momentum, we know that the five ball will have a final velocity directed at 60° below the positive x-axis in our coordinates. With this information in hand, we can simply write the equations of momentum conservation for our system.
Recall that we are assuming external impulses are negligible compared to the internal impulse of the collision.
\begin
[ p^
_
+ p^
_
= p^
_
+ p^
_
]
[ p^
_
+ p^
_
= p^
_
+ p^
_
]\end
Using the fact that the cue ball and 5-ball have equal masses, and substituting any known zeros, this becomes:
\begin
[ v^
_
= v^
_
+v^
_
]
[0 = v^
_
+ v^
_
]\end
We can rewrite these equations using geometry:
\begin
[ v^
_
= v^
_
\cos(\theta^
)+v^
_
\cos(\theta^
[v^
_
\sin(\theta^
) = v^
_
\sin(\theta^
) ]\end
We can now solve for each final speed in turn, finding:
\begin
[v^
_
= \frac{v^
_{i}}{\cos(\theta^
) + \cos(\theta^
)\sin(\theta^
)/\sin(\theta^
[ v^
_
= \frac{v^
_{i}}{\cos(\theta^
) + \cos(\theta^
)\sin(\theta^
)/\sin(\theta^
)}]\end
These expressions are already fairly simple, but they can be made even simpler by realizing that since:
\begin
[ \theta^
+\theta^
= 90^
]\end
we know:
\begin
[ |\sin(\theta^
)|=|\cos(\theta^
)| ] [|\cos(\theta^
)|=|\sin(\theta^
)|]\end
Using this plus the fact that sin2 +cos2 = 1, it is possible to simplify our answers to the form:
\begin
[v^
_
= v^
_
\cos(\theta^
) = \mbox
]
[ v^
_
= v^
_
\sin(\theta^
)= \mbox
] \end
Can you see that this result satisfies the elastic collision requirement (remembering that the masses are equal)? Considering the assignment of the cosine and the sine, we see that as the cue ball's angle grows the speed it retains shrinks. Does this make sense?
Part B
The cue ball is moving at 2.0 m/s when it impacts the five ball, which is at rest prior to the collision. The cue ball exits the (perfectly elastic) collision with a speed of 1.5 m/s. What angle with respect to the original direction of travel of the cue ball is made by each ball after the collision?
Solution
System, Interactions and Models: As in Part A.
Approach: