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# Momentum WA: Billiard Ball Problem

## Question

I'm hoping this is the right place to post for Physics C Problems.

I thought I heard that people were confused by the billiards web assign so I thought I'd weigh in. To me it seems like the velocities of the cue ball and the 8-ball have been switched on the webassign, but maybe I'm crazy. Also, I only know from experience that perfectly spherical billiard balls that collide perfectly tangentially ricochet off each other at complimentary angles, but the mathematical explanation in the answer packet leaves a bit to be explained for me. Any clarification would be much appreciated!

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Some thoughts:

Start off with conservation of energy to relate the velocities, so that:

$v_{ci}^2 = v_{cf}^2 + v_{8ball}^2$

Note that this is a Pythagorean Theorem relationship, therefore they must form a right triangle, and the angles between $v_{cf}$ and $v_{8ball}$ must be 90°. You can use this to determine your angles after the collision. A diagram here is helpful.

Next, use conservation of momentum in the x-direction to relate all x-component velocities before and after the collision:

$m{v_{ci}} = m{v_{cf}}\cos {\theta _{cf}} + m{v_{8ball}}\cos {\theta _{8ball}}$

You can also use conservation of momentum in the y-direction in the same way:

$0 = m{v_{cf}}\sin {\theta _{cf}} + m{v_{8ball}}\sin {\theta _{8ball}}$

Solve these equations simultaneously to obtain $v_{cf}$ and $v_{8ball}$.

Reminder: Your calculator must be in degree mode!

:egg)

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