I was watching a video; the amount of collisions made between two colliding blocks on a frictionless surface with walls on either side that have infinite mass always equals the digits of pi. So say one block is 100kg (block A) and the other is 1kg (block B) and we are assuming perfectly elastic collisions, the total amount of collisions before block Ahits the opposite wall would be 31. We keep increasing block A’s mass and the numbers go up: ex. 314 at 10 000kg, 3141 at 1 000 000kg.
After that video I tried to understand *why* this happens but I am no mathematics expert. If we’re being honest here “elastic collisions” was a stretch for me haha I had to reach back to my high school physics memories.
In: 323
Maybe a visual could help.
Get a piece of paper and a pencil and draw a horizontal line and above it a square. This is a top view of the block approaching the wall. From the edge of the block closest to the wall, draw a line to the wall, then back to the block. This is the movement the block makes as it hits the wall and bounces away. Just a simple line, down then up.
Now, to the right of that drawing, I want you to do that same down then up motion, but move your hand to the right as you do it. If you end up with a lot of V’s, Try to make it a fluid motion as you do, and you’ll end up with a wiggly line. This is a called a wave and it repeats as it moves right. One up and down movement is 1 oscillation.
Now, somewhere else on the paper, going from left to right, draw the bottom half of a circle. Then, without taking your pencil from the paper, draw a top half of a circle while still moving to the right. Should look a bit like an S on its side. Notice that this looks pretty much the same as the wave you drew a second ago. That’s because they are the same.
Therefore, all waves are just circles and anything that defines a circle, defines a wave. Therefore 1 oscillation of a wave is equal to the circumference of a circle, which we know is 2(pi)r, where r is the radius of the circle. Therefore the block’s movement is related to pi. Also, make a note that the big block also moves in 1 oscillation, it moves left until it stops then moves back to the right.
Now there is a math equation that explains how two objects interact when they hit each other called “the Conservation of Momentum” Which is the speed and weight of one object relating to the speed and weight of another object. We are controlling some variables, but the whole Elastic thing is that the overall speed is always the same, therefore if the big block slows down, the small block must move faster and if the big block speeds up, the small block must slow down.
Since we are controlling a lot of variables here, this equation and the equation of a circle basically are the same. Which means that the speeds of two objects as they collide are also related to pi. So, since the speeds and the movement are related to pi and pi describes circles, you can draw the interaction between the objects as a circle.
Now each time the block hits the big block or the wall, a portion of a circle is drawn and each portion of the circle has to be equal because the overall speed never changes. The mass of the objects in relation to each other defines how big each portion of the circle is. The bigger the block, the smaller the portion. Then it becomes how many portions can be drawn without going past 1 full circle.
At this point we get into trigonometry to explain how big each portion is based on the size of the two blocks, and you can watch the video to explain how the math is resolved, but long story short, the square root of the big block multiplied by pi gives you the number of collisions. So if the big block is 100, then it becomes 10*pi, which is 31.41, and since we can’t have half a collision, we have 31 collisions.
Or in terms of drawing the circle, you can draw 31.41 portions and if you drew the 32 portions, you would have more than 1 full circle, which would mean you have more speed than you started with, which can’t happen.
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