# How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls?

Consider the following scenario. Let $\alpha>1$. Suppose whenever two superelastic balls collide at speed $\gamma$ they bounce off each other at speed $\gamma\cdot\alpha$ (i.e. $\alpha$ is the coefficient of restitution). One uses the center of mass frame of reference in order to determine the speed of the balls after they bounce so that the laws of Newtonian physics are satisfied.

In this scenario, since $\alpha>1$, two billiard balls gain kinetic energy after they bounce off each other. Suppose furthermore for simplicity that each superelastic ball is completely frictionless, each superelastic ball has the same mass, and that no force shall act on any of the balls except for the balls bouncing off each other.

Suppose now that one has a collection of $n$ superelastic billiard balls of diameter $1$ randomly placed in a region $U$ of $\mathbb{R}^{d}$. Initially the super bouncy billiard balls have velocity $0$. However, one stray billiard ball is launched from $\infty$ towards the point $\mathbf{0}$ with speed $1$ and the now $n+1$ balls form a huge explosion. Approximately what is the expected value (and also the distribution if that can be calculated) of the amount of energy released by the explosion?

For example, if $f_{d}^{\alpha}(r)$ is the expected value of the kinetic energy after the explosion where $n=r^{d}$ and $U=\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|<r\}$, then how about fast does the function $f_{d}^{\alpha}$ grow? I would imagine that the function $f_{d}^{\alpha}$ would exhibit exponential growth.

If the dimension $d=1$, then one may want to assume that the balls are simply points with 0 diameter. If $d>1$, then I would imagine that eventually the balls will stop bouncing into each other and hence after the explosion the amount of energy will be finite. However, if $d=1$, then it is unclear whether the balls will necessarily ever stop bouncing off each other.

• This reminds me of the following question: mathoverflow.net/questions/156263/… – André Henriques Oct 25 '15 at 21:57
• In the statement of the problem, is the rule $\gamma\rightarrow\gamma\alpha$ supposed to apply in some fixed and arbitrarily chosen frame of reference, or in the center of mass frame? If the former, then the motivation for the question seems weak, since it would violate Galilean relativity. – Ben Crowell Oct 25 '15 at 23:07
• Ben Crowell. Yes. One should use the center of mass frame. After all, after each bounce, the amount of kinetic energy should increase so that we have an explosion. And yes, all the laws of Newtonian physics except having coefficient of restitution greater than $1$ should be satisfied. – Joseph Van Name Oct 25 '15 at 23:36
• @JosephVanName: OK, could you edit the question to clarify that? BTW, there is nothing about $\alpha>1$ that violates Newton's laws. – Ben Crowell Oct 26 '15 at 0:18