Baloo

This may serve as nice fodder for cosmological arguments (by drawing an analogy between the "First cause" and the "Last bounce"). Very cool problem (and solution) - thanks, guys!

simian

One of the calculus problems I had in college was a shape (bouned by two lines where y-> +/- infinity as x ->0, y->0 as x -> infinity. Finite volume, infinite surface area - it could be filled with paint, but there would not be enough paint to actually cover the surface area (if looked at as a 3-D object).

Simian

Lobstrosity

Right, I remember something like that, simian. I got to thinking and I figured out what it was:

take the curve y = 1/x where x >= 1 and create a surface by rotating the curve about the x axis. Alternatively, you could define this surface and the volume it encloses easily in cylindrical coordinates by the expression 0 <= r <= 1/z, z >= 1.

Basically, it looks like a trumpet or horn (if I recall, it was called Devil's Trumpet or something like that...maybe) whose large end is the unit circle and whose narrow end is infinitely small and resides out at infinity.

So let's look at the volume of this guy. Since it's just comprised of a bunch of stacked circles of varying radii, you can get the volume by summing the areas of all these circles:

V = integral from 1 to infinity of pi r² dz
= integral from 1 to infinity of pi/z² dz
= -pi/infinity + pi/1 = pi

We see therefore that the volume is finite and happens to be exactly pi.

Now let's look at the surface area. Once again, since it is just stacked circles, you can get the surface area by summing the circumphrences of all these circles:

S = integral from 1 to infinity of 2 pi r dz
= integral from 1 to infinity of 2 pi/z dz
= 2 pi ln(infinity) - 2 pi ln(1) = infinity - 0 = infinity

Thus the surface area is infinite, which seems hard to believe given that it encloses a finite volume. As simian says, you can fill it with paint (say pi cubic meters of paint, if our axes are in meters), but you would need an infinite amount of paint if you wanted to cover the surface.

As an interesting aside, if the bounding surface curve decayed more sharply (as y = 1/x², for example), then both the volume and the surface area would be finite.

SULPHUR

theoretically it will never stop. If graphically plotted using time and distance it becomes it becomes assymptonic whi ch gives a value of infinity

Lobstrosity

Please read the rest of the thread. This is nothing more than Zeno's paradox retold. There is a time when the ball is not bouncing.

Abacus

Theoretically it is stopped after ~2.61 seconds.

SULPHUR

I assume vou took the same idea as simian, that when time and distance became infinitely small they were neglible and need not be part of the problem

Jesse

It's not that time or space become negligible, it's that an infinite series of bounces can fit into a finite time, because each successive bounce takes less and less time. In the same way, the infinite sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2.

SULPHUR

Why is there finite time? Where is finite time first mentioned?

Lobstrosity

Finite time isn't mentioned anywhere, it's something you calculate when you actually solve the problem. Add up the time for each bounce. Though there are an infinite number of bounces, the time decreases per bounce in such a way that the series converges. Let's use the example Jesse gave, but let's be more specific. Let's say I have a ball that bounces such that each bounce takes half as long as the previous. If the first bounce takes 1 second, for how long does it bounce? Well, the total bounce time is:

T = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

This is an infinite series. It tells you that the ball bounces an infinite number of times. However, this series does not sum to infinity--it converges. Specifically, it converges to 2 seconds (i.e. T = 2). The ball will bounce for two seconds, after which it will not be bouncing at all (and we are not making any approximations about bounces being negligible at this point). All infinity bounces occur during those two seconds.

As I said, this is just another formulation of Zeno's paradox. You might actually find it interesting for it demonstrates how we can become unreasonably hung up when faced with infinities.