TITLE: Perfectly centered break of a perfectly aligned pool ball rack
QUESTION [95 upvotes]: Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a triangle) has every ball equally spaced apart and all balls touching all other appropriate balls. All balls are perfectly round. Now, imagine that the cue ball was hit along a friction free surface on the center axis for this triangle O-------<| and hits the far left ball of the rack dead center on this axis. How would the rack react? I would imagine this would be an extension of newtons cradle and only the 5 balls on the far end would move at all. But in what way would they move? Thanks

REPLY [18 votes]: This is more of a comment inspired by Jim Belk's answer than an answer to the question in itself. It is also more physics than mathematics. However, I hope I can help readers see that this system is related to some very interesting science.
I wanted to say a little bit about the dynamics of the energy transfer in the billiard break. Very naïvely, you might think of a triangular lattice of interacting particles (the billiard rack) as acting like a discrete model for a crystalline solid. Solids support sound waves (just knock on your table and listen), and so again naïvely we might expect that the billiard break occurs via some strong compressional sound waves propagating through the system, akin to those vibrations induced when knocking on wood.  However, this naïve intuition is wrong!
A system of billiards all just barely touching, like Newton's cradle, is an example of a sonic vacuum (see e.g. this expository physics article). 
The term was coined by Vitali Nesterenko to describe mechanical systems where the speed of sound vanishes and the tiniest perturbation must lead to supersonic shocks -- behavior very far from that of ordinary sound.  The usual formula for the speed of sound in say a chain of identical masses joined by identical springs is $v_s=\sqrt{k/m}a$ where $k$ is the spring constant, $m$ is the mass, and $a$ is the length of the springs. In a system interacting via a nonlinear force law like Hertz's, the effective spring constant is $df/d\delta|_{\delta=0}=0$ (where $\delta$ is the overlap between particles) which actually vanishes! Thus the speed of sound is zero.  This is a very special situation which occurs because the particles are all just barely touching.
The consequence of all this is that energy cannot propagate via ordinary sound, and what appears instead is a shock wave, satisfying a nonlinear PDE. In Jim Belk's pictures, this is what we see as the "shock front" colored yellow traveling from the point of impact through the billiards. 
Nesterenko wrote down a continuum limit for systems of Hertzian particles (the following is specialized to particles arranged on a triangular lattice): 
$\frac{\partial^2\xi}{\partial t^2}=c^2\frac{\partial^2}{\partial x^2}\left[\xi^{3/2}+\frac{2R^2}{5}\xi^{1/4}\frac{\partial^2}{\partial x^2}(\xi^{5/4})\right]$
(If one wants to study shock waves in systems of particles interacting with other force laws, then some of the exponents here change). Also, none of the above discussion really depends on the "perfectly centered" initial condition. I think the paper I cite below may give you some idea of what can happen as we vary the initial angle.

It might be interesting to compare the solutions of this equation with suitable boundary conditions to simulations along the line of Jim Belk's for a very large pool ball rack (large enough at least so that one can follow the shock wave for an appreciable length of time).

The derivation of the above equation is apparently in Nesterenko's book "Dynamics of Heterogeneous Materials", though I learned it from this very relevant paper coauthored by some friends of mine: Transmission and reflection of strongly nonlinear solitary waves at granular interfaces, by 
A. M. Tichler, L. R. Gomez, N. Upadhyaya, X. Campman, V. F. Nesterenko, V. Vitelli.  In this paper they study the propagation of shocks through interfaces between packings of billiard balls of two different masses. I think you'll agree it's quite close to the billiard break.
Here's figure 2 of Tichler et al's paper.