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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 cueball and the 9-ball have mass m , and the cue ball is hit at an initial speed of v_i. Jeanette carefully hits the cue ball into the 9-ball off center, so that when theballs collide, they move away from each other at the same angle theta from the direction in which the cue ball was originally traveling (see figure). Furthermore,after the collision, the cue ball moves away at speed v_f, while the 9-ball moves at speed v_9.

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 theta through which the 9-ball travels away from the horizontal, as shown in the figure. Perhaps surprisingly, you should be able to obtain anexpression that is independent of any of the given variables!

Express your answer in degrees to three significant figures. theta = rm degrees

Relevant knowledge:

Complementary angles – Any two angles whose sum equals 90° is called the complementary angle.

If A and B are the two complementary angles then-

∠A+∠B=90°

In the right triangle, the two angles are not related to one another.

Supplementary Angles – Two angles that have a sum of 180° can be called supplementary angles.

If X and Y are the two supplementary angles then-

∠X+∠Y=180°

The opposite angles of a rectangle can be complementary.

This problem can be set up knowing it is elastic. Conservation of momentum and conservation kineticenergy will be observed.

So, you get:

p

_{I}= P_{F}mvi = MVF + MV9

Divide by mass to get: vi = Vf + v9

To get isolated equations you could use trig substitutions for each x or y componentofmomentum, but I found it easier to go on to conservation ofkinetic energy.

You now have kinetic energy.

K

_{I}=K_{f}1/2 mvi2 = 1/2 mvf2 +1/2 mv92

Divide by 1/2 mass again: vi2=vf2+v92

This should look exactly like the Pythagorean Theorem. The tail to head method of vectors will also produce an image with a 90 degree angle. 2nd = 90 =>

θ =45

^{o}.