Constructing the digits of pi from conservation of energy and momentum in collisions

A short one, just to get the ol’ bloggin’ fingers movin’ again’. It’s starting to get chilly outdoors and they’re simply covered in cobwebs!! Where did they come from? How did I get spider webs on my hands…..?

Last year, 3Blue1Brown produced a series of lovely videos (but really, they all are) on a strange little thing. An object is sent bouncing back and forth between a rigid wall and another moving object with a particular (integer!) mass ratio. If this ratio is a power of 100, the number of collisions will be some of the first digits of pi!! Isn’t that nuts.

My two blocks.

Anyone who has taken a high school senior level physics class or above would recognize the need to implement two conservation laws: that of momentum (linear with velocities) and that of energy (quadratic with speeds). Someone with just a bit of programming experience should be able to set up a series of solutions for the speeds after successive collisions, until the “no-collide” condition is met: the two blocks are traveling in the same direction, away from the wall, and gaining distance between them.

Hey, I’ve got a bit of programming experience! I whipped this up to see if I could get the same results.

Both blocks equal. The block on the right is 100^o = 1 times more massive than the block on the left. Three collisions.
100:1. There are 31 collisions in there. Looking good!

10000:1. There are 314 collisions! It looks like it’s working. Trust me. You must trust me.

It looks like they’re doing it right! A fun little exercise. In the spirit of the original prompt I’ll let you find or figure out why it works for yourself. You can always watch Sanderson’s explanation video.

One hint I can give is it is a bit related to the Buffon’s needle technique of finding pi. This is a fun idea that understanding could open some avenues of thought for more difficult estimation problems. Check out the description and explanation on Numberphile: