SRL Archive - Mathematics

back in highschool, in one of the advanced classes lead by university professors, one evening the guy brought in a bouncing ball, one of those that bounces very high because it preserves energy, they act as nearly ideal bounce balls, i.e. almost all energy of impact is preserved and minimum is lost on heat due to deformation

he drew the following on the blackboard


with following explanation:

in both pictures we discount the effect of gravity,

in the first picture we see an infinitessimally small object with ideal bounce, and no mass, we throw it towards the floor at 45 degrees, it will bounce up at 45 degrees and hit the bottom of desk, then it will bounce on again at 45 degrees

in real life however, the ball will have mass and will have non-zero diameter

as you can see in second picture, if you throw it at 45 degrees, it will also bounce at 45 degrees off the floor, but where will it go after hitting the bottom of the desk



needless to say that the answer was not as obvious and so we wrote the equations for angular momentum and energy preservation, noting that the ball touches the floor not at the center but at its very edge

the results were unexpected but simple bouncing ball and a desk proved that the results were correct.

where will the ball go?

A random guess, but I'm saying a 90 degree angle?

I'm just visualizing it skidding across …

at the point of first impact, as pictured, the rotational momentum will become counter clockwise so at the point of second impact it will actually do the opposite of skidding, for the ball to skid, at the point of second impact, the rotation would have to be clockwise

So it will bounce back to where it bounced?

"the ball touches the floor not at the center but at its very edge"

what do you mean?

It bounces back towards you due to the spin on the ball, try it if you don't believe me…

basically that is the case - it goes right back to where it came from, but it's more fun when you write down the equations, solve them then go whaaaaaat? and then try it with the ball afterwards :)

If the ball doesn't hit the ground in the center each time, how are all those bounces 45 degrees?

exactly - it doesnt

so when you calculate it using energy preservation, on first hit, it bounces off at 45 degrees but it acquires counterclockwise spin equivalent to the 90 degree change in direction, so when you apply the 90 degree to the next bounce, it ends up being 180 degree bounce and ball follows back from under the table towards the spot on the floor that it bounced off previously

the ??? is 135 degrees (i.e. 45+90)

It bounces rotating clockwise on the first impact but only on it's radius aka the edge.. nothing every touches the center of the ball. On the first bounce, it transfers some energy to the floor and the next bounce, some to the table in which the momentum changed, and it rolls counter-clockwise now. the last and final bounce will not be in the exact same place as before but it will be close enough and this is where the ball will lose more energy to the floor eventually bouncing around back and forth enough until all the energy is lost and it lays there still. P.S. By lost I mean transferred.

Too lazy to do the calculations but we also did this in Grade 12 physics :c same with the rubberband experiments :c The worst topic for me was the light through a medium and light through a slit with a specified width.. my god :c