Bouncing Ball

[PlanetJeroen]

Shhh .. did you hear that ? Those where crashing neurons on fried receptors. The playtime of this video is misleading .. I swear it takes half an hour at least !

(Nice one tho, thanks for helping me get it too :))

[sisyphusorianus9787]

There's nothing more amusing than having a discussion on newtonian laws and ontology disrupted by the cheerful, adroitly inconspicuous sounds of an ice cream truck!

(2:59)

[holbrookprieto]

ha ha... yes i can hear the truck...!

[JoakimfromAnka]

I like this one even if I only understand a fraction of it. Moar strange philosophical topics please! :D

[TheHonestTheist]

PICKLES!!!!

er...what are we doing again?

[rozeboosje]

Pardon me?

You just stroked another person's ego based on the premise that they responded to you "respectfully". You might like to consider doing the same.

[TheHonestTheist]

damn dude, it was a joke after seeing the other person say "ICE CREAM!!!!" lighten up

[rozeboosje]

You first

[peanarchist]

ICE CREAM!!!!

[rozeboosje]

So do I

[shorty108764]

Are you some kind of mathematician or something? What are you trying to prove by this?

[rozeboosje]

I have some mathematical grounding. I'm not trying to prove anything, I'm just casting doubt on the bald assertion that infinite regression is "impossible". That means that certain arguments are invalid. You can't draw further conclusions from that, I'm afraid, but I hope it might make you think.

[yupiie]

I liked what you did there.

[rozeboosje]

Thanks!

[theStickerBush]

As I start to raise my hand for a question, Roze slaps my head... so I guess I will stop there.

The ball theoretically never reach its goal depending on how you look at it. This is the Achilles and tortoise fable, right? Neither complete the race because no one can ever reach there goal. Hence the infinity argument... am I following you correctly?

[rozeboosje]

Not quite, but sort of related. Achilles and the tortoise also constitutes an infinite sum: 1/10 + 1/100 + 1/1000 .... or something of that nature. The important realisation is that by adding an infinite number of small terms together you may arrive at a FINITE result. In the Achilles example, 1/10 + 1/100 etc = 0.1111111.... and of course that's just 1/9.

The point is that they DO reach their goal even though intuition may suggest otherwise. The problem is that intuition is often wrong.

[theStickerBush]

Your a stud, thanks for the comparisons. I understand clearly.

"The problem is that intuition is often wrong."-I agree

[TheReasonWhyGuy]

lol

twice the height was my guess.

[rozeboosje]

:-)

[LesPaulLiberty]

Very interesting, and nice. It's sure making my head spin, thinking about other 'infinite' things (the universe, hell *though I don't believe in it). 5 stars

[rozeboosje]

Cheers!

[desertlight]

A way to reconcile the mathematical description with the physical reality is to realize that as the bounce height approaches an infinitesimal value, the surface that it falls to is also moving, even if just Brownian movement. Effectively, the surface moves and touches the ball, resulting in transfers of energy outside the original equation. For all practical purposes the mathematically described bouncing stopped, but, the ball will never stop moving as long as there is available energy.

[desertlight]

The end result might be thought of as Newtonian Mechanics meets Quantum Mechanics. Another way to think of the results is that when (i) is large the value added by the addtional terms is negligible and the distance between the asymptotic value and the convergent value is essentially zero. An additional infinite number of terms wouldn't add anything significant. At this point the surface is touching the ball and any motion is independent of the orginal mathematical description.

[rozeboosje]

Yeah. But even if you make this a perfectly Newtonian and ideal situation, with a totally immovable surface and a perfect ball, the maths shows us the ball will still stop bouncing, after a finite amount of time, and after having executed an infinite number of bounces.

[desertlight]

True, but, wouldn't it be fun to be small in the idealized world with the perfect ball having exactly 0.50 elasticity and watch what happens?(I feel a song coming on...) I've been looking for massless inextensible string for so long that I've given up and gone skiing on the local frictionless inclined plane.

[rozeboosje]

LOL. Yeah!

[dQuarters]

I'm missing the point... BUT:

I know the infinite bouncing was a condition of the initial equation, but I don't understand how you can't then work the sum backwards to count the bounces. Can you?

How could that second reversed equation = infinity?

Also, how could something infinite have a 'starting' point?

math + me = 0

[rozeboosje]

You can't count the bounces, because there are an infinite number of them. But you can add up the distance covered, and the amount of time spent bouncing, and even though those are arrived by adding an infinite number of terms to each other, the total values are finite. And then you get those counter-intuitive facts, like an infinite number of bounces starting a finite amount of time in the past

[dQuarters]

What I meant to ask is if you only know the following:

-the ball was dropped at 10 meters

-it bounced in halves from that point

-it bounced a distance of 30 meters

-it bounced for 8.24 seconds

Does the magic ball still bounce infinitely?

I'm assuming my reverse logic is somehow moot here. But I don't understand the math of my own equation (let alone yours), and I'm hoping you can shed a little light on my confusion so I can watch part 2 with clarity.

Sorry to pester.

[rozeboosje]

Yes, it would bounce an infinite number of times (not infinitely) - it's implied by the second point "it bounced in halves from that point". The distance of 30 meters, too, is implied, but the 8.24 seconds would present a bit of a problem as it is derived from the acceleration due to Newtonian gravity. It wouldn't be impossible, I guess.

[dQuarters]

Well now that's just nutty. I'll have to assume you and your magic balls make mathematical sense and carry on.

Thanks.

[rozeboosje]

LOL Magic balls eh? :P

[InfinitelyManic]

Please explain your distinction between "infinite number of times" and "not infinitely". If the ball "starts" bouncing then are you merely suggesting that the cardinality of all subsequent infinite bounces are less than the cardinality of all bounces of a ball that was always bouncing?

[rozeboosje]

No. "infinitely" means "for an infinite *amount* of time". An infinite number of times does not need to add up to an infinite amount of time. As illustrated.

[1classified1]

the whole point is, that there are an infinite amount of bounces WITHIN the time in which it takes to come to complete rest. A converging series will have an infinite amount of terms even though it sums to a finite amount. Here each term is the equivalent of one bounce, and the sum is the total time.

what i want to know is though, what actually happens in real life?

[InfinitelyManic]

Now comes an infinite number of questions. What is REAL life?

[1classified1]

i mean, when you actually bounce a ball, what happens to it? In theory it bounces an infinite amount of times, but what happens in practice?

That was my question.

[InfinitelyManic]

In practice (i.e., from our physical perspective), the ball stops bouncing. I also think, in practice, that an infinte number of tasks cannot be completed in a finite amount of time.

[1classified1]

but you can think of a single smooth motion made up of an infinite amount of infinitesimally small, quantised motions, so why not?

[InfinitelyManic]

Ok. How about moving the cursor of a slide rule, on the A scale, from 1 to 10, which can be done in a finite amount of time. Since there are an infinite amount of rational numbers, among other things, between 1 and 10, are you suggesting that this rises to the level of completing an infinite number of tasks in a finite amount of time?

[1classified1]

yes. infinity can be wrapped up in finite quantities.

For example the 3D shape "Gabriel's Horn" has a volume of Pi units, but an infinite surface area.

Similarly fractals are 2D shapes with a finite area but infinite perimeter.

It is therefore perfectly acceptable for an infinite number of tasks to occur in a finite time.

before you complete the task of reaching 10cm, you must reach 5cm. but before that you must reach 2.5cm, and before that 1.25cm and so on.

[InfinitelyManic]

Very good and fascinating points; however, we can't, for example, physically construct Gabriel's horn [in real life]. So are we somehow unknowingly making the argument that "a cat has nine tails" by moving back and forth between physical and the imaginative dimensions?

[1classified1]

we cant construct them, but they do exist.

And where has this cat argument come from?

[InfinitelyManic]

The Cat Has Nine Tails argument is just a fallacious argument wherein the definition of a word(s) is changed to make the argument seem true.

1) You would agree that no cat has 8 tails? Yes

2) You would agree that one cat has got one more tail than no cat? Yes

Therefore, one cat, has one more tail than no cat (which has 8 tails) and so it has 9 tails.

[desertlight]

I found this interesting. It seems related to the notion of integration where an infinite partitioning of area under a curve still sums to a finite value. The only difference seems to be that the width of the partitions are unequal but diminishing toward zero as the number of them increases without bound. I haven't thought about the concept of taking limits for 40 years. If I remember right, the infinite series for P would be called a convergent series.

[rozeboosje]

I'm not good on the technical terms but that sounds right, yes.

[desertlight]

Another fun notion is the non-linearity of the times between bounces, allowing an infinite number of bounces within a finite time. Listening to a coin twisting as it falls and oscillates on a flat surface I can hear the periods decreasing quickly. If one walks one-half the distance to a wall and repeats by one-half and so on, then can one reach the wall if one can also increase one's velocity toward infinity, thereby, reducing the time period of each travel segment toward zero?

[0092b]

I tried to do the math in this vid, but the dutch educational system has failed me. =(

[rozeboosje]

Hm. How long ago did you go through that? I'm Dutch myself. I was taught this in or around the early '80s.

[0092b]

Ik begrijp hoe je de som opbouwt en omzet naar je antwoord. Ik heb alleen nooit geleerd te rekenen met het teken dat je gebruikt om de oneindige reeks gebeurtenissen (2*5 + 2*2,5 + 2*1,25 + ...) samen te vatten, als je begrijpt wat ik bedoel.

Ik heb HAVO gedaan en ben vorig jaar geslaagd =P met een 6,5 voor wiskunde.

[rozeboosje]

Prima! Tja. Wat kan ik zeggen? De enige keer waar ik het ooit nog gebruik is in discussies zoals deze dus het is niet echt iets wat je dagelijks nodig hebt. :P

[niceonetom]

i understand how the ball's distance travelled asymptotically approaches a finite value, but you stated in the premise that it will NEVER stop bouncing. how can you then calculate a value for when it will stop?

or does the ball bounce an infinite number of times in a finite number of seconds? i.e the individual bounces approaching being infinitely close together in time? what a mind fuck.

[rozeboosje]

The latter. It bounces an infinite number of times within a finite amount of time. And that then adds up to a finite total. Crazy but true.

[Eldxale]

Half of n cannot equal zero unless n is already zero.

[rozeboosje]

True, but surely I didn't claim that anywhere in this video, did I?

[Eldxale]

You claimed it would stop bouncing. How is that not claiming that 2 * 0 != 0?

[Eldxale]

This bouncing ball will never go further than infinitely close to twice its first bounce, so if its first bounce is 10 m, it will never reach 20 m. But, since n/2 can't equal 0 unless n is already 0, it will keep bouncing forever.

[rozeboosje]

It's the sum that counts. You can add an infinite number of positive values together and arrive at a finite result. That is the underlying mathematical understanding behind this video. Intuitively your mind will baulk at this, but the maths adds up nicely and it is correct.

[rozeboosje]

Yes, it will stop bouncing. But there won't be such a thing as a "last bounce".

[InfinitelyManic]

In the instant case, the geometric series (the bouncing ball summation formula) converges and thereby has a limit-as it approaches infinite. This helps us to know when the ball "appears" to stop bouncing (if you allow for an imaginative infinitesimal distance between bounces).

[rozeboosje]

No. "appears" does not come into the question. As the limit of the *number* of bounces approaches infinity, you'll see that the total amount of *time* spent bouncing approaches a finite value. It does not exceed that value, and as a result, if you look at the ball *after* that point in time, it will no longer be bouncing, infinitesimally or otherwise. It has stopped bouncing.

[InfinitelyManic]

I thought your point in the video was merely imaginative. At 4:53 you say, ". . . even though the ball bounces an infinite number of times . . ." then your text reply says, "It has stopped bouncing." This appears to conflict at first blush. Please resolve.

[rozeboosje]

I present the maths in the video, if I remember correctly.

[Tiwaking]

Interesting mathematical take on Zeno's paradox.

[bubonicnate]

ICE CREAM!!!!!!!!!!!!!!!!!!!

[rozeboosje]

What I never mentioned was that I rounded a off as equal to 10. The real value on earth is somewhere around 9.8 m/s/s

[paulconcepcion]

I've never seen magic expressed as a mathematical equation before. :-)

I am neither a physicist, nor a mathematician, and I do not claim to fully understand your argument. (Perhaps I missed the point entirely). But intuition is based on observation, rather than mathematical formulae; therefore, I must dismiss magic as the explanation.

[rozeboosje]

Definitely. I only described the ball as "magic" because I wanted to stress the fact that I am not describing a real ball in the real world, but just a mathematical abstraction.