We’ve had a lot of threads about the digits of pi. Do we have room for yet another one? Everybody knows you can calculate pi by dropping toothpicks on wood-planked floor, but I stumbled across webpages describing another physical experiment which yields pi. Indeed, it’s one of the simplest possible experiments in simple mechanics.
Here’s the experiment: Line up three objects with masses ∞, 1, M. The object with mass ∞ is a rigid wall; (the 1 is 1 kilogram but units won’t matter). M is the only variable in the experiment. Initially all three objects are stationary, and you start by pushing object M toward object 1 (give M velocity 1, the units don’t matter). There’s no gravity, no friction; all collisions are elastic. Only one dimension (x) matters, y = z = dy = dz = 0 for all objects. Start the clock when M bumps into 1. What happens? How many collisions in total will there be?
When M is tiny, it will bounce back off of 1 as a first collision; and 1 will move slowly and eventually collide with the wall. TWO collisions total.
When M = ⅓ the initial collision will give 1 a velocity of ½ and M a velocity of -½. This is the way to conserve both momentum
. . . 1·½ - ⅓·½ = ⅓·1
and kinetic energy
. . . 1·½[sup]2[/sup] + ⅓·½[sup]2[/sup] = ⅓·1[sup]2[/sup]
After 1 bounces off the wall, it will have the same velocity as M. Or slightly more velocity if M is slightly more than ⅓.
So if M=.3334 there will be THREE collisions compared with TWO for M < .3333.
Solve for M=1? A very elegant case has THREE collisions; M=1.0001 would yield FOUR collisions. Here are some more results:
M = 10 –> 10 collisions
M = 100 –> 31 collisions
M = 1000 –> 99 collisions
M = 10000 –> 314 collisions
M = 100000 –> 993 collisions
M = 1000000 –> 3141 collisions
Notice anything? Here are some more examples:
M = 100000000 –> 31415 collisions
M = 10000000000 –> 314159 collisions
M = 1000000000000 –> 3141592 collisions
M = 100000000000000 –> 31415926 collisions
The digits of pi! Although I do realize that “arithmetic is what it is”, I still found this astounding!
Here (pdf) is an elegant proof of this relationship “PLAYING POOL WITH π”.
Here’s someone who thinks this experiment may help us understand quantum search! “Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search”
