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.
