Let the subscripts c and 9 represent the cue and the 9 balls, respectively.

From the law of conservation of kinetic energy:

$\begin{array}{rcl}{\mathbf{K}}_{\mathbf{i}}& \mathbf{=}& {\mathbf{K}}_{\mathbf{f}}\\ \frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}_{\mathbf{c}}{{\mathbf{v}}_{\mathbf{i}}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}_{\mathbf{9}}{{\mathbf{v}}_{\mathbf{i}}}^{\mathbf{2}}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}_{\mathbf{c}}{{\mathbf{v}}_{\mathbf{f}}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}_{\mathbf{9}}{{\mathbf{v}}_{\mathbf{f}}}^{\mathbf{2}}\end{array}$

Jeanette is playing in a 9-ball pool tournament. She will win if she sinks the 9-ball from the final rack, so she needs to line up her shot precisely. Both the cue ball and the 9-ball have mass,and the cue ball is hit at an initial speed of . Jeanette carefully hits the cue ball into the 9-ball off center, so that when the balls collide, they move away from each other at the same angle from the direction in which the cue ball was originally traveling (see figure). Furthermore, after the collision, the cue ball moves away at speed , while the 9-ball moves at speed .

For the purposes of this problem, assume that the collision is perfectly elastic, neglect friction, and ignore the spinning of the balls.

Find the angle that the 9-ball travels away from the horizontal, as shown in the figure.

Express your answer in degrees to three significant figures.

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Elastic Collisions concept. You can view video lessons to learn Elastic Collisions. Or if you need more Elastic Collisions practice, you can also practice Elastic Collisions practice problems.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Menary's class at York University.