## Elastic collisions and $\pi$

Posted on: January 17th, 2013 by
1

Here is a surprising relation between elastic collisions and $\pi$:

Start with a large ball of mass $16*100^N$ kg. on the very left, a small ball of mass $1$ kg. to it's right and a wall to the very right. Send the large ball towards the small ball with an initial velocity $u_0$. The small ball will keep bouncing back and forth between the large ball and the wall. Assume all collisions are perfectly elastic. Eventually the large ball changes direction and starts moving away from the wall. What's surprising is that the number of collisions between the large ball and the small ball before the large ball changes direction is given by the first $N+1$ digits of $\pi$. As an example, if $N=2$, the large ball is $16*100^N=160,000 kg.$. In this case, the large ball undergoes 314 collisions before it changes direction. For $N=5$, it undergoes 314,159 collisions before changing directions. This result is exact (assuming perfectly elastic collisions and perfect mass without any errors). It's explained very well in the following youtube video: