Weird ways to calculate pi professional#

We can calculate the velocities of the masses after each collision by making use of conservation of energy, conservation of momentum for the two mass collisions and velocity reversal at the wall, as follows: A beautiful visual simulation of the process can be found on 3Blue1Brown’s YouTube channel. This is a ridiculously inefficient way to calculate of course, but the pattern was first published by G. If you don’t believe this, I don’t blame you! I didn’t myself when I first came across it, until I had done some numerical calculations to convince myself of the connection. These are the digits of, and the pattern continues! For, then the number of collisions will be given by the first digits of !

Note the pattern in the number of collisions: 3, 31, 314, 3141. If is times the mass of then will experience collisions. Repeat with times the mass of and will undergo collisions. This time will undergo 314 collisions in total. Again there will be multiple collisions before moves away to negative infinity faster than. Mass will undergo 31 collisions before this happens.Īgain we repeat the process, but with now ( ) times the mass of. In fact will bounce between the wall and many times before it has not only slowed to a stop, but also imparted enough momentum to get moving to the left again at a speed greater than its own. After bounces off the wall it will collide with, slowing it a little more, but still not stopping it, while itself shooting off towards the wall again. In this case will again move off towards the wall, but will not come to a complete stop – it will continue moving to the right at a slightly reduced speed. Suppose we repeat the previous scenario with mass now 100 times that of. There will have been three collisions in total. Now mass will stop and mass will move off to negative infinity. Mass will now collide with the wall and return with the same speed (velocity will have opposite sign) to collide again with mass. If the two masses are equal then when they collide, mass will stop and transfer its momentum to mass, which will move to the right with unit velocity. We will assume mass is initially stationary ( ) a short distance from the wall, and mass is moving towards it from the left with unit velocity ( ).

When mass collides with the wall momentum is not conserved the velocity of the mass is simply reversed. When the two blocks collide with each other momentum is conserved during the collision. Figure 1: Sliding blocksĪll collisions are elastic and there is no friction, and so energy is conserved at all times. When mass hits the wall at the right it rebounds elastically. More specifically, consider elastic collisions between two blocks of mass and that slide without friction, as illustrated in Figure 1.

Seeing this brought to mind a weird way of calculating the digits of by counting the collisions between sliding blocks.

Mathematics Teacher Training ScholarshipĪt the time of writing, the roads and pavements are icy and people are slipping and sliding all over the place.

Weird ways to calculate pi professional#

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