If a ball bounces back to half its height every time it bounces. How can it ever come to rest?

Question

Mind ****!

Answer 1

While it is entirely true that a ball which bounces to half its height each time will never come to rest, there DOES come a point where it doesn't matter.

As an illustration, there was a professor at the University who had a beautiful daughter studying physics. Two students, one a mathematician and the other an engineer wanted to marry her. Unable to decide which was the more suitable mate, she asked her father to propose a contest.

Each suitor would stand at opposite ends of the football field with the daughter on the 50 yard line. The father would blow a whistle from the announcers booth, once a minute. At each whistle, each boy would advance half the remaining distance to the girl. The first boy to reach her would win her love.

Upon hearing the rules, the mathematician immediately realized that an infinte number of steps and time would never allow him to reach the girl. He could not believe that she would propose such a foolish test, and walked off the field in disgust.

The engineer and the girl both calmly waited for the first whistle, knowing that in six short minutes he would be close enough to kiss his prize.

The moral of the story is that while the limit may never reach zero, there is still a point only 7 steps away that is "close enough" to zero as to be practically indistinguishable from zero.

Answer 2

This is a restatement of Xeno's Paradox which states that if a hare runs twice as fast as a tortoise, and both start running at the same time with the toroise X meters in front of the hare, after a certain interval of time, the distance between them will be halved. After another interval, the distance is halved again. No matter how many intervals you take, there is always half the previous distance remaining, therefore the hare never catches the tortoise.

As the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. There may be an infinite number of intervals, but they add up to a finite number.

This is not the same as the bouncing ball, since the ball would bounce forever; however, when the distance drops to quantum dimensions, strange things begin to happen, and the ball will stop. it will not take an infinite time to reach, because of the decreasing intervals between bounces as in the resolution of Xeno's Paradox.

Answer 3

Actually, this is a materials question. I haven't done physics in years, but I can try to explain it; if the ball is made of rubber, a certain amount of energy expended from its being dropped is stored and "fired" back the other way, causing it to bounce. Rubber is good for this as it is compressible. However, the quantity of energy to make rubber compress and bounce enough to lifts its own weight in Earth's gravitational field is NOT zero. For this reason there is a definite point where the ball will be unable to bounce off the ground as it has passed beneath this point. The ball would only keep bouncing for ever if it had a mass of 0 and the second law of thermodynamics is conveniently ignored.

Answer 4

If a ball really did bounce back to half its height every time it bounced it would never completely come to rest. That is half-life - it would keep halfing, never getting to zero, although it would become inperceptible. Theoretically what you'd end up with is a ball vibrating off the ground very slightly.

Of course in reality that simply doesn't happen. The ball just lands on the floor eventually.

Answer 5

If a ball were to bounce half its height every bounce and not lose any energy except that which will make it bounce half its height then it would, in theory, bounce infinitely. But unfortunately we are not in an ideal magical World so it loses energy to its surroundings via heat energy, wind resistance, friction in the floor etc...

Answer 6

This is a rephrasing of one of Zeno's Paradoxes (if you fire an arrow at a running person it takes time for the arrow to reach them but by that time the person has already moved on so the arrow needs to travel further to reach the point they have moved to, but they have already moved from that point to, and so on and so on.) If you look at it logically the ball will never stop bouncing, it will just bounce to smaller and smaller heights, but in real life, not strictly bound by logic this doesn't happen.

There are some mathmatical proofs for this, but I don't really understand them, you can find them at http://en.wikipedia.org/wiki/Xeno%27s_paradox#Proposed_solutions

Answer 7

Because it loses energy each time it hits the ground, upon some final collision with the ground it stays there.

However, if you walk 1/2 the distance to a wall, stop, then 1/2 the distance are at that point and so on, do you ever reach the wall?

Answer 8

This is an instance of Zeno's paradox. It will not take long for the rebound height to become so small that it is indistinguishable from thermal motion, and at that point you can say that it has effectively stopped.

Answer 9

When it goes sailing through a neighbor's window. You may be absolutely certain it will come to rest. But then YOU will begin to bounce, back and forth with your neighbor, until the whole thing is resolved. - Chris.

Answer 10

Friction