Question from Linear Kinetics

Question

A billiard player wishes to place his ball as close as possible to a pocket without falling into it. The two balls have a mass of 165 g each. The deceleration of the ball on the table is 0.1 m/s2, and the angle between the resulting balls’ directions (v1’ and v2’) is 90°.

a) Calculate the initial velocity of the white ball, just after being hit by the cue.

b) Calculate the impulse produced by the cue.

Answer 1

Hi there Infinity!

To start off, typically we like to convert our mass to units of kilograms so we have m1=m2=165g=0.165kg

We are given that the deceleration of the ball (with better etiquette negative acceleration), is 0.1m/s².

Let's assign our cue ball to be mass 1 (m1) and have velocity 1 (v1), and the white ball (m2, v2). (Strange choice of color coding since a cue ball is the white ball?)

Because this is an elastic collision, denoted by the context of billiard balls (since they just bounce off of each other rather than sticking together), we can assume that there is a conservation of momentum.

The equation of momentum for a system like this is Momentum = Mass * Velocity or p=mv.

We also know due to the law of conservation of momentum, that our initial momentums will be equivalent to our final momentums (since we are calculating them before the table begins to impose a deceleration).

So: p1initial + p2initial = p1final + p2final: m1v1i+m2v2i=m1v1f+m2v2f. Since all the masses are equal to each other, we can divide it out to be left with v1i+v2i=v1f+v2f.

We can assume that our initial velocity of the white ball is equal to 0 since it is stationary then bringing our equation to v1i=v1f+v2f.

Despite verbiage saying initial velocity of the white ball for part a, in our equation it is v2f.

Part a) The Initial Velocity of the White Ball after being hit by the cue ball in an elastic collision is equal to

the initial velocity of the cue ball minus the final velocity post collision of the cue ball or (v1i-v1f).

Impulse is equal to the change in momentum of an object. In this case our mass stays the same and so we have ΔP=mΔv. Since our change in velocity is from 0 to (v1i-v1f), we can say that our impulse is equal to m(v1i-v1f) or

Part B) 0.165kg * (v1i-v1f).

This problem is tricky because it presents you with excess information that may convolute your process. Just stick to the basic equations and figure out what you actually need.

Let me know if you need any clarification!