Why doesn't the 9th ball move in the break in the nine-ball pool game?

In the game of nine-ball pool, we break the rack like shown below: In the break, we hit the 1st ball with the cue ball. Many people familiar with pool games say that if the rack is constructed properly (i.e. there're no gaps between adjacent balls), the 9th ball won't move a bit.

As far as I've seen, this folklore is very true. Usually, the 9th ball moves little.

I've googled much and failed to reach any mathematical or physical proofs about this. Does anyone know whether this is true?

• Assume a hit on the line through the centres of the 1, 9 and 8 ball for simplicity. By the symmetry all actions and reactions on the 9 ball cancel out. Same principle as for the Newton's cradle. – Phoenix87 Jan 23 '16 at 11:47
• Thanks for your comment. I know about the Newton's cradle, but I think the situation is more complicated since we have to think about multiple-body collisions, while Newton's cradle includes only two-body collisions. – Kota Ishihara Jan 23 '16 at 12:06
• Newton's cradle can have as many bodies as one likes. – Phoenix87 Jan 23 '16 at 12:26
• That's right, but when we discuss the dynamics Newton's cradle, we can break it down into multiple two-body collisions. In contrast, in this problem, we have to think about three-body collisions. For example, ball 1 hits 2 and 3 at the same time and this process cannot be decomposed into two two-body collisions. – Kota Ishihara Jan 23 '16 at 12:47

The non-linearity referenced above is due to the Hertz ($\alpha x^{\frac{3}{2}}$) stress-strain relationship for solid spheres. The pool problem, unlike Newton's cradle, also has friction between the balls and the table. The soliton pulses, nevertheless, are due to the elastic deformations of the balls and pass before the balls begin to roll. Friction does play a role in permitting the change of direction of the soliton pulse at balls 4 and 5 (effectively yielding a reflection). Once the balls are moving due to their unbalanced momenta, friction will cause rolling rather than sliding.