My immediate guess agreed with my mathematics. But I have the advantage that a variant of this problem had occurred to me some 20 years ago, and I worked it out.
In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely. There may be some residual vibration that is not apparent, but it sure seems to act like this toy model of the situation says it should.
In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely.
Indeed, I've done that -- but my intuition told me that it was stopping due to friction, not due to the exponential decay of an infinite number of bounces. :-)
It's oddly disappointing to realize that the model actually acts more or less the same as the real world for once.
Indeed, I've done that -- but my intuition told me that it was stopping due to friction, not due to the exponential decay of an infinite number of bounces. :-)
But your intuition was correct! Without friction the bounces would be perfectly elastic and wouldn't go into exponential decay! :-)
In fact, pick up a bouncy ball and drop it on a flat surface. You can observe the fact that bounces become smaller and more rapid, and then they stop bouncing entirely. There may be some residual vibration that is not apparent, but it sure seems to act like this toy model of the situation says it should.