When does the ball stop bouncing?

[Philosophy Explorer]

Take a rubber ball and let it drop to the ground. On its first fall, it takes 1 second. It bounces upwards and then on its second fall, it takes 1/2 second to reach the ground so it takes a total of 1.5 seconds to reach the ground after the first bounce. After the 2nd bounce, it only takes 1/4 second to reach the ground so after two bounces, it takes a total of 1.75 seconds to reach the ground.

This pattern continues whereby the time after each successive bounce to reach the ground is half the time from the previous bounce so if it took 1/2 second to reach the ground after the first bounce, then it must have taken a full second to reach the ground before the first bounce. Here's the question. How long will it take the ball
to complete all of its bounces?

In math it says the total time will be two seconds even though the ball goes through an infinite number of bounces. The reason is that the time between bounces gets progressively less. Some people will say the ball keeps on bouncing forever due to the infinity aspect.

What do you think?

PhilX

[wtf]

I think you're recycling your old material from another forum.

Have you any original thoughts?

[Philosophy Explorer]

I think you're recycling your old material from another forum.

Have you any original thoughts?

No original thoughts at the moment.

For those who don't know, Zeno's paradoxes has inspired this thread. Infinity and its implications are always fascinating to me. Maybe the question is what does infinity mean to you?

PhilX

[wtf]

No original thoughts at the moment.

The snark just writes itself. I'll pass on the opportunity.

Maybe the question is what does infinity mean to you?

Or maybe the question is, what specific questions do you have remaining after the three page thread here ... http://onlinephilosophyclub.com/forums/ ... 12&t=10314

After that thread, what issues are still confusing to you? Or are you just recycling your old material because even you have gotten bored of your own schtick.

[Philosophy Explorer]

No original thoughts at the moment.

The snark just writes itself. I'll pass on the opportunity.

Maybe the question is what does infinity mean to you?

Or maybe the question is, what specific questions do you have remaining after the three page thread here ... http://onlinephilosophyclub.com/forums/ ... 12&t=10314

After that thread, what issues are still confusing to you? Or are you just recycling your old material because even you have gotten bored of your own schtick.

If I were bored with it (and I was aware I did it already), I wouldn't recycle it.

Also I'll mention something a mod had mentioned to me here. I'm not concerned with what goes on with other forums.

PhilX

[wtf]

Also I'll mention something a mod had mentioned to me here. I'm not concerned with what goes on with other forums.

I'm not privy to that conversation. Nor do I post from the privy. I already posted a perfectly sensible response to the bouncing ball problem (probably under another handle) on that other forum.

I'm seriously asking you what you still don't understand about the bouncing ball after three pages of conversation on that other thread. The situation was perfectly well broken down for you both physically and mathematically. You know that mathematically, given perfect elasticity, no friction, and a continuum model of spacetime, the ball bounces forever. And physically, energy loss from friction and the Planck-scale limits on distance and time cause the ball to eventually stop.

And you were already told these things last year.

So what else do you need to know?

[Philosophy Explorer]

Also I'll mention something a mod had mentioned to me here. I'm not concerned with what goes on with other forums.

I'm not privy to that conversation. Nor do I post from the privy. I already posted a perfectly sensible response to the bouncing ball problem (probably under another handle) on that other forum.

I'm seriously asking you what you still don't understand about the bouncing ball after three pages of conversation on that other thread. The situation was perfectly well broken down for you both physically and mathematically. You know that mathematically, given perfect elasticity, no friction, and a continuum model of spacetime, the ball bounces forever. And physically, energy loss from friction and the Planck-scale limits on distance and time cause the ball to eventually stop.

And you were already told these things last year.

So what else do you need to know?

No. Even mathematically the ball may stop if successive bounces (as is the case here) get sufficiently reduced as time moves on (if the series were a harmonic series instead of powers of two, then the ball would continue to bounce).

You're under the misimpression that my threads are for me. They're for all users to consider.

PhilX

[wtf]

You're under the misimpression that my threads are for me. They're for all users to consider.

Can't speak for others. They may find your warmed-over misunderstandings more interesting than I do. Please do note that I don't often give you a hard time for posting your questions. But for posting yesterday's stale bread, yes that needs to be called out. People can comment or not, but at least they'll know what they're commenting on.

[Philosophy Explorer]

You're under the misimpression that my threads are for me. They're for all users to consider.

Can't speak for others. They may find your warmed-over misunderstandings more interesting than I do. Please do note that I don't often give you a hard time for posting your questions. But for posting yesterday's stale bread, yes that needs to be called out.

Two more things. First I presume you and HalfWit are the same. Second you say it's impossible in a physical (real) universe for the ball to stop bouncing which I disagree with - it depends on how the bounces progress from one step to the next.

Just because you were on the other forum doesn't mean much to me. I'm only currently concerned with this one

PhilX

[wtf]

Just because you were on the other forum doesn't mean much to me. I'm only currently concerned with this one

PhilX

Ok you're right. Forget all that. You raised an interesting point about what the physical ball does if the bounces are 1/2, 1/4, 1/8, etc., and you claim that the ball (does? doesn't) stop in that scenario. I wasn't entirely clear what your intention is because above you said the opposite of what I thought you were trying to say. So just remind me, the harmonic series makes it go forever but the 1/2^n series makes it stop? Or the other way 'round?

I don't think I can get to this tonight but I'll stipulate that the other thread was irrelevant and once I clearly understand your point, I'll try to engage with it.

[Philosophy Explorer]

Just because you were on the other forum doesn't mean much to me. I'm only currently concerned with this one

PhilX

Ok you're right. Forget all that. You raised an interesting point about what the physical ball does if the bounces are 1/2, 1/4, 1/8, etc., and you claim that the ball (does? doesn't) stop in that scenario. I wasn't entirely clear what your intention is because above you said the opposite of what I thought you were trying to say. So just remind me, the harmonic series makes it go forever but the 1/2^n series makes it stop? Or the other way 'round?

I don't think I can get to this tonight but I'll stipulate that the other thread was irrelevant and once I clearly understand your point, I'll try to engage with it.

It is the harmonic series that makes it go on forever. The
1/2ⁿ series (or more generally the 1/kⁿ series where k is 2 or greater) is the one where the ball stops.

PhilX

[wtf]

It is the harmonic series that makes it go on forever. The
1/2ⁿ series (or more generally the 1/kⁿ series where k is 2 or greater) is the one where the ball stops.

Ok got it. Can you please remind me of the particulars of the thought experiment? Does it bounce to the same height every time but faster and faster? Or is the height proportional to the time, so the successive heights are the same as the successive times?

Also in the physical version, are there units? Is the height measured in feet or meters or whatever? Reason I ask is that given the unit of length, it will be easy to calculate the particular bounce that represents a time or a distance smaller than the Planck scale. At which point your argument will fall apart. Not to get ahead of myself, but that's where this is going. That's the problem with all supertask type thought experiments. They're not physically realizable.

[wtf]

Ok let's "do the math" as they say.

The Plank length is around 1.6 x 10^(-35) meters.

[Moderators, any chance of MathJax on this forum? It's a small block of code in the HTTP header, brain-dead easy to implement].

Now, nobody really knows what the Planck length means. It might be the actual mesh size of a discrete universe. Or, it might just be a limit on what we can know. In other words the essential nature of the Planck length might be ontological, or it might be espistemological. Nobody knows.

But from the Wiki article:

The Planck length is believed to be the shortest meaningful length, the limiting distance below which the very notions of space and length cease to exist.

So we really can't talk meaningfully about any distance smaller than the Planck length. Either it doesn't exist in the world; or we can never know anything about the world at a smaller scale. Either way, we can't talk sensibly about it.

Let's say the first bounce is from 1 meter up; the second is to 1/2 a meter, and so forth. How many bounces does it take before we can no longer claim we know what we're talking about? No peaking, make a guess.

Let's work it out. We seek n such that

2^n = 10^(-35)

I'll ignore the 1.6, it doesn't make any significant difference.

Take the natural log of both sides and use the log laws:

n x ln(2) = (-35) x ln(10)

or

n = -35 x ln(10) / ln(2)

From Wolfram Alpha we find that n = -116 or so. So after 116 or so bounces, you are claiming a physical significance for a distance that has no meaningful physical significance.

What say you? How do you justify your thought experiment as a physical idea? After 116 bounces, you are below the scale where physical law is known to apply.

https://www.wolframalpha.com/input/?i=- ... %2F(log+2)

Every single supertask-type idea breaks down at the Planck length. Also we could do the same calculation with the Planck time. After a certain number of bounces, you can not meaningfully say that the bounce is taking 1/2^whatever seconds. The laws of physics do not support that conclusion.

If you claim a physical result, you need to respect the laws of physics. That's why supertasks don't work.

By the way, your idea of using the harmonic series 1/2 + 1/3 + 1/4 + 1/5 + ... also fails, and for the exact same reason. Once the n-th term goes below the Planck length or the Planck time, your argument can't be applied.

[Philosophy Explorer]

Ok let's "do the math" as they say.

The Plank length is around 1.6 x 10^(-35) meters.

[Moderators, any chance of MathJax on this forum? It's a small block of code in the HTTP header, brain-dead easy to implement].

Now, nobody really knows what the Planck length means. It might be the actual mesh size of a discrete universe. Or, it might just be a limit on what we can know. In other words the essential nature of the Planck length might be ontological, or it might be espistemological. Nobody knows.

But from the Wiki article:

The Planck length is believed to be the shortest meaningful length, the limiting distance below which the very notions of space and length cease to exist.

So we really can't talk meaningfully about any distance smaller than the Planck length. Either it doesn't exist in the world; or we can never know anything about the world at a smaller scale. Either way, we can't talk sensibly about it.

Let's say the first bounce is from 1 meter up; the second is to 1/2 a meter, and so forth. How many bounces does it take before we can no longer claim we know what we're talking about? No peaking, make a guess.

Let's work it out. We seek n such that

2^n = 10^(-35)

I'll ignore the 1.6, it doesn't make any significant difference.

Take the natural log of both sides and use the log laws:

n x ln(2) = (-35) x ln(10)

or

n = -35 x ln(10) / ln(2)

From Wolfram Alpha we find that n = -116 or so. So after 116 or so bounces, you are claiming a physical significance for a distance that has no meaningful physical significance.

What say you? How do you justify your thought experiment as a physical idea? After 116 bounces, you are below the scale where physical law is known to apply.

https://www.wolframalpha.com/input/?i=- ... %2F(log+2)

Note that the way I've set up the problem, the Planck length doesn't enter the picture - it's irrelevant to the problem.

You asked:

"Can you please remind me of the particulars of the thought experiment? Does it bounce to the same height every time but faster and faster? Or is the height proportional to the time, so the successive heights are the same as the successive times?"

Proportional.

You also asked:

"Also in the physical version, are there units?"

No units. That's why Planck's length is irrelevant.

PhilX

[wtf]

No units. That's why Planck's length is irrelevant.

PhilX

What do you mean no units? You are proposing a physical thought experiment. A ball falls through a distance. After some number of bounces, the distance ceases to be physically meaningful.

You drop the ball from a height of 1. In the physical world, there has to be a unit. Whether it's one meter, one mile, a million miles, or the edge of the known universe, it's a finite length. After a certain number of bounces, your experiment ceases to be physically meaningful.

You need to explain how the laws of physics become irrelevant in a thought experiment about physical law!