Question

You are shooting pool with some friends of yours and are beautifully positioned to win the game by knocking the 8-ball into a corner pocket. In fact, all you need to do is strike the 8-ball head-on---it's just an ideal situation. Assuming you execute the shot, how fast will the cue ball be moving after hitting the 8-ball? (assume that the masses of all the pool balls are basically equal).

How fast will the 8-ball move as a factor of the initial speed of the cue ball? (that is, what's the ratio of the 8-balls final speed to the cue ball's initial speed?)

Answer 1

For an ideal situation, we can assume that it is an elastic collision, it means that the momentum and energy must be conserved.

Conservation of Momentum:

where: m_c and m₈ are the masses of the cue ball and the 8-ball respectively, v_ci and v_8i the initial velocities of the cue ball and the 8-ball respectively, v_cf and v_8f - the final velocities of the cue ball and the 8-ball respectively.

Conservation of Kinetic Energy:

and now, from the givens m_c and m₈ are equal to each other, m_c = m₈ = m and the 8-ball is at rest, it means that its initial velocity is equal to zero: v_8i = 0.

We have two unknowns: v_cf and v_8f and 2 equations, it means that we can solve this system:

and after simplifying it, we get:

and we plug it into the equation we got from the Conservation of Kinetic Energy after simlifying the masses:

and we get:

and from this equation we have several situations:

or

And now we can answer the questions:

The cue ball will be not moving after hitting the 8-ball: v_cf = 0

The 8-ball will be moving with the same velocity as the cue ball had before the collision: v_8f = v_ci and the ratio of the 8-ball's final speed to the cue ball