Height and Distance

Sounds like a job for a mathematical series.

Can you write a partial sum that would suggest a series?

Infinite.

No, I think it's bounded. But calculus was a very long time ago...

How do you find the solution?

Well, the way the problem is stated it is a theoretical situation. It does not take into considerations the elasticity of the ball and assumes each bounce returns 80% of the initial energy. In reality there are other forces that soon stop the ball from bouncing, but stated as it was, this is an infinite series.

In other words, the problem is one where we approach a limit of zero for the hight of the next bounce. Since each bounce is 80% of the previous bounce we approach an asymptote, but never reach zero. Thus the answer is infinite because 80% of something is never zero.

Think of it this way. A pretty girl is standing at the other end of the room. Every second you move 80% closer. Will you ever reach the girl? No, but you will get close.

Ah, Zeno's paradox!

I don't want to publish an answer...I'll shoot you an e-mail.

Thus the answer is infinite because 80% of something is never zero.

I assume you mean, by "the answer" that there is an infinite number of elements in the series. "The answer" the poster is seeking, though, is the accumulated distance... which has a definite value, although that distance is approached asymptotically as time wears on (with the ball causing annoying noise at first, but soon going above the limits of hearing, and not too long after giving off microwaves, etc.)

Mathematically you will find the theoretical answer to this question when you do a series but.....

In the real world we have losses. Put these losses in this system and you come to a finite number of cycles.

Short enough to do a physical test perhaps?

Do three or so tests at one given height and plot the amount of bounces. That way you can guestimate the real values. Do this for several heights and plot as well. Draw a fitting curve and extract your answer to predict all amount of bounces as a dependend on height. As long as you stay on earth you should be ok.

Once you know the number you can do a simple sum.

Bob is your uncle

I'm assuming the ball is dropped from height H, and bounces to .8H on the first bounce, to .64H on the second, to .512 on the third, etc.

I've known engineers who use Excel (or Lotus in the old days) for everything, including as a word processor. I don't use it as a word processor, but for problems like this it is simple to do a series of copy/paste operations and from so doing you will see what the answer will end up being. There is a clear single answer: in other words, there is a number which will not be exceeded and if you do about 40 calculations of bounce height, you will see what that number is.

Whether you then consider the total to be the up and down travel, or just the travel down, depends on question interpretation. Of course, then there is the physicist's answer, based on change in potential energy answer: H.

As other posters have said, in practice it won't be exactly like solution below, but I guess you want the theoretical answer. This is straightforward, it's a geometrical progression with an infinite number of terms, and it doesn't sum to infinity, even in theory. Only thing need to be careful of is, the first drop H only happens once, the subsequent heights happen twice.

So distance = H + 2*0.8*H + 2*0.8²*H + 2*0.8³*H........

= 2*H*(1 + 0.8 + 0.8² + 0.8³......) - H = 2*H/(1 - 0.8) - H = 2*H/0.2 - H = 10*H - H

= 9*H.

Cheers....Codey

Precisely. Or imprecisely, depending upon your philosophy. If you start your bouncing, and come back in a year, the ball will still be bouncing, and the sum will not yet have reached 9.000... On the other hand, it will have reached 9.0 in a surprising short time, and, assuming H = 1 meter, the bounces will have dropped to nanometer size (and smaller) in fairly short period too.