This is something I dreampt up in a physics lab about imagining infinity but I never got round to modeling it. Seems like it could get out of hand pretty quickly!
Imagine we have a ball in a 1d system that bounces back and forth between point x1 and x2. Every time the ball hits either x1 or x2 it reverses direction and doubles speed.
After time t what will be the number of times that x1 will be hit.
Given initial conditions of x2-x1 = 1m and initial postion = x1 and initial speed 1 m/s, how many times will x1 be hit after 5s, 10s, and 20s.