Think of this plot in terms of area. The area on the graph covered by the blue data (to hit the ball 2) is much larger than the area on the graph showing the speed needed to hit the ball 3. If you go *a lot* more difficult to get a collision involving all four balls.

Let’s do one more. What if I add a 4 ball to the collision chain?

Just to be clear, this is a comparison of the initial cue-ball speed range that results in ball 3 hitting ball 4. Let me go into some big intervals for the initial cue-ball speeds.

To make ball 1 hit ball 2, the speed x could be from close to 0 m / s to 1 m / s. (I did not calculate speeds greater than 1 m / s.) The speed y could be from about 0.02 to 0.18 m / s. This is a speed range x of 1 m / s and a speed range of about 0.16 m / s.

To have ball 2 hit ball 3, the speed x could be from 0.39 to 1 m / s with the speed y from 0.07 to 0.15 m / s. Note that the speed range x has dropped to 0.61 m / s and the speed range y is now 0.08 m / s.

Finally, for ball 3 to hit ball 4, the speed x could be from 0.42 to 1 m / s and the speed y from 0.08 to 0.14 m / s. This gives an x range of 0.58 m / s and a y range of 0.06 m / s.

I think you can see the trend: More collisions mean a smaller range of initial values that will result in success in the final ball.

Now we need to prove the final case: *nine* balls. Here’s what it looks like:

OK, that works. But will the last ball always be hit if we do it in an extra gravitational force caused by the interaction between the cue ball and the player?

This is pretty easy to prove. All I have to do is add some kind of human. I use a approximation of a spherical human. So, people are not really spheres. But if you want to calculate the gravitational force due to a real player, you would have to do some seriously complicated calculations. Each part of the person has a different mass and would be a different distance (and direction) from the ball. But assuming that the person is a sphere, then it would be the same as if the whole mass were concentrated in a single point. *This* it is a calculation we can make. And in the end, the difference in gravitational force between a real and spherical person probably doesn’t matter too much.