A question I need answered.

[midnghtraver]

First of all.

Lets make sure it is known that you can cut 1 in half, then in half again, then in half again infinity times, no?

Second of all.

Lets make sure it is known that you can walk half way to something, then half way again, then half way again, infinity times. You would just end up going miiiccrrooo fractions of length.

So here is the question:

If I drop a bouncy ball. How does it hit the ground and come back up?

If I drop it from 3 feet high




It will drop to 1.5 feet


then .75 feet

then .375 feet and so on.

So how does the ball ever reach 0 (the table) and bounce back up!?

When does the denominator of 1/infinity ever break infinity and go to zero!?

[iironiic]

Theoretically, it continues to bounce. However in practice (due to air resistance or friction, etc), the ball stops bouncing which is completely understandable.

Also infinity is not a number; it's a concept. To be mathematically accurate, instead of 1/infinity, say the lim as x --> inf of 1/x, which in fact equals 0

[Izzy]

My only guess is that once the force is less than that needed to go against gravity it just stops.

Originally Posted by iironiic

To be mathematically accurate, instead of 1/infinity, say the lim as x --> inf of 1/x, which in fact equals 0

The limit of 1/x as x approaches infinity is said to be zero, but that doesn't mean that it actually gets to zero.

[midnghtraver]

So what your saying is the table isn't 0 because it isn't in terms of the table its in terms of gravity?

EDIT: And the reverse of gravity via table is not 0 but in between 3 feet and x --> inf of 1/x

[MarioNintendo]

Lmao, I remember asking this question to myself a long time ago. My guess was that once gravity was pulling the ball to the table more than the bounce of the ball itself, it would just stop moving. I'm not sure, though. One thing is certain, this is no maths problem, it's a physics problem.

[midnghtraver]

Originally Posted by MarioNintendo

Lmao, I remember asking this question to myself a long time ago. My guess was that once gravity was pulling More than the bounce of the ball, it would just stop. I'm not sure, though. One thing is certain, this is no maths problem, it's a physics problem.

One of which my physics teacher couldn't answer >.<

I'm not asking how does the ball stop or the bounce of the ball or any of it. Just the fact that it has to hit the table to bounce up and that based on the "halfing rule" it should never be able to touch the table.

[Izzy]

Originally Posted by MarioNintendo

Lmao, I remember asking this question to myself a long time ago. My guess was that once gravity was pulling the ball to the table more than the bounce of the ball itself, it would just stop moving. I'm not sure, though. One thing is certain, this is no maths problem, it's a physics problem.

Physics is math.

Originally Posted by midnghtraver

One of which my physics teacher couldn't answer >.<

I'm not asking how does the ball stop or the bounce of the ball or any of it. Just the fact that it has to hit the table to bounce up and that based on the "halfing rule" it should never be able to touch the table.

Now I have no idea what you are talking about. What reasons does a ball dropped above a table have to not hit the table? It hits the table because thats just what happens when two objects move towards each other with no interference.

[midnghtraver]

Originally Posted by Izzy

Physics is math.




Now I have no idea what you are talking about. What reasons does a ball dropped above a table have to not hit the table? It hits the table because thats just what happens when two objects move towards each other with no interference.

But that contradicts the halfing rule.

[devonin]

He's asking you to solve Zeno's paradox.

If anything moving from point A to point C must past through midpoint B, it is logically impossible to ever reach C, becuase there is always a midpoint B which must be passed through.

Once you get to midpoint B, it becomes point A, and you have a new midpoint B to reach on your way to C.

Philosophically, the problem with presenting this as a paradox to be solved is that there are a few things you have to establish first. Yes I grant your claims at the start about infinitely divisible lengths. Fair enough. Now, prove that Time functions in the same way. Show me an actual discrete unit of time that isn't purely arbitrary, and then divide that in half for me.

Mathematically, there's already a solution inherant in the concept of the convergent series. It is actually the case mathematically that if you add together the reciprocals of the powers of two (which is what you're doing in this paradox -> 1/1 + 1/2 + 1/4 + 1/6....) your result is "2"

[Izzy]

If you understand what hes asking devonin can you explain it to me in a way that makes it sound more like a problem?

All I am reading is "why does the ball hit the table when dropped?" I don't even see where the halving things comes into play.

Edit: I looked up zeno's paradoxes and the one in question seems more like a play on words. It is saying that there is an infinite number of half points between 2 points so how can you ever complete an infinite number of moves. Things just don't work that way so who cares? There is always an infinite number of points between 2 points but obviously you can walk around. Personally I wouldn't even consider this a paradox or a problem at all.

[midnghtraver]

Originally Posted by Izzy

If you understand what hes asking devonin can you explain it to me in a way that makes it sound more like a problem?

All I am reading is "why does the ball hit the table when dropped?" I don't even see where the halving things comes into play.

Edit: I looked up zeno's paradoxes and the one in question seems more like a play on words. It is saying that there is an infinite number of half points between 2 points so how can you ever complete an infinite number of moves. Things just don't work that way so who cares? There is always an infinite number of points between 2 points but obviously you can walk around. Personally I wouldn't even consider this a paradox or a problem at all.

Yes and no. I didn't know this was labeled haha. And Devonin pretty much answered it. The ball on the table was a physical example so you would understand it easier. I don't care why the ball hits the table. I wanted to know in terms of my halfing rule how it was possible TO hit the table, when it obviously did because it bounces back up.

[reuben_tate]

I think the problem is that you are thinking of the position of the ball as a function of x where f(x) is the ball's position relative to the table, x is the time elapsed and f(x) = 3*(1/2)^(x) (this is assuming that the position is halved every one second). However, most objects don't move like that. In this case, the position of that bouncy ball relative to that table can be expressed as f(x) = 3 - 9.8(x^2) [when (0<=x<=(sqrt(15)/7))].

A better example to question is whether radioactive materials ever do decay completely, since the function of their remaining mass relative to time, never reaches zero, although it does approach it.

I have a feeling I could have worded the above a bit better, but whatever.

[qqwref]

According to modern physics, in real life, everything is quantized, which means that everything, even time and distance and energy, has a smallest possible amount. So what you think of as a second is really some whole number of the smallest-possible-time unit. It's so small, though, that you never notice this effect. But if you divide a second in half, and then in half again, and so on, sooner or later you will get down to that level. When you have one unit, you can't divide it anymore, because time really doesn't exist at a more detailed level.

In the ball bouncing example, there's also dissipation of energy involved. Each time it bounces, the ball has to waste energy deforming, waste energy making a noise when it hits the table, and waste energy pushing air molecules out of the way. Because energy is also quantized, this ends up meaning that the ball can't bounce an infinite number of times, because once the remaining energy in the ball is small enough, all the remaining energy will be used up, and it won't continue to bounce.

Originally Posted by reuben_tate

A better example to question is whether radioactive materials ever do decay completely, since the function of their remaining mass relative to time, never reaches zero, although it does approach it.

Radioactive material is a bit tricky. You start with a certain number of radioactive atoms, and you know that - on average - after each half-life about half of them will decay. Or, each one has a 50% chance of decaying after one half-life. Chances are, though, it won't be exactly one half; if you start with 100 atoms, maybe after one half-life you'll have 40 left, and then you'll have 20, and then 10, and then 6. Who knows. And when you only have one atom, if you wait one half-life, there's a 50% chance that there is still one atom, and a 50% chance that it's gone (decayed).

So it's not true that the number of remaining radioactive atoms never approaches zero - it will eventually, but you don't know how long you will have to wait. It's technically possible that they will all decay before even one half-life... and also possible that the last atom will take billions of years to finally give up. You can think of the atoms like popcorn kernels popping - the closest you can get to "everything is popped" is "only one is left unpopped".

[Reach]

Mathematically, there's already a solution inherant in the concept of the convergent series. It is actually the case mathematically that if you add together the reciprocals of the powers of two (which is what you're doing in this paradox -> 1/1 + 1/2 + 1/4 + 1/6....) your result is "2"

It's a good mathematical answer. It's problematic though and doesn't actually answer the question.

The philosophical objection I see to this solution is that, since the sum of 1/1 + 1/2+ 1/4 + 1/8 only = 2 at infinity, and since we live in a quantitative world where the ball can only move at a finite velocity, the ball should still never hit the ground, because it can never move at any given point an infinite number of infinitely small distances. Accordingly, the ball would have to indefinitely remain an infinitely small distance away from the ground.

Obviously this is wrong. If I drop a ball, it hits the ground. But why?



The most basic solution to this is along the lines of what qqwref said. At a quantum level, the essence of what space time is begins to break down at a planck length. That is, distances or points between that of a planck length do not actually exist**.

This immediately resolves the paradox. Without an infinite number of points within points, there are a finite number of distances between the ball and the ground. Therefore, the ball must inevitably hit the ground.


So, the solution is in a way a fundamental objection to the initial premise. Mentally or mathematically you can cut 1 in half infinitely, but physically you cannot.

Analogously, you could cut an imaginary pie in half indefinitely, but once you've cut it into individual atoms and separated those atoms into respective protons, and into respective gluons and quarks, you cannot cut it in half anymore. Doing so would destroy the precious pie completely, much like moving the final piece of distance between the ball and the floor causes the ball to hit the floor.


**Note: When I say that it doesn't exist, there is some philosophical trickery here. Physically, in terms of reality itself being composed of matter and energy, there is literally nothing continuous beyond the plank length. Points within points at a plank length simply do not physically exist in a continuous way like the rest of the universe as we perceive it does. The universe at this scale can essentially be thought to exist as granules that themselves do not have any spatial extent.

[ledwix]

Also, at some point along its future of bouncing up at ever-shrinking heights, the bounce height becomes so small that the electrostatic forces become much more prevalent than anything provided by gravity, and so it becomes meaningless to talk about, for instance, a "bouncing" iron atom, because the bounce height becomes small compared to the diameter of the atom. At some point, it has to go under the threshold of allowable quantized energies and become nonexistent. Plus, the electric potential would be much greater than gravitational.

Nature has to be discontinuous at that level. Continuity is a mathematical convenience, and it is very well approximated in the physical world. But it is not rigorously true. I guess everyone has said that already, though.

[Reincarnate]

This is an example of Zeno's Paradox. It's basically a way of asking "if we can divide some finite distance into infinitely many pieces, then how can we possible make it from one end to the other? There are INFINITELY many pieces to traverse!"

It's not really a paradox -- the question just sounds confusing because of the way it's phrased and because people tend to have a hard time grasping infinity. In this case, the fact that you're talking about a finite distance already solves the problem.

Say we drop the ball 10 feet. To traverse 10 feet, you have to first traverse 5. But to traverse that, you need to travel 2.5 feet, and so forth. Ultimately we can keep going until we ask ourselves the question, "How can anything move at all? To travel some distance X we have to travel some distance smaller than X first, but to travel that we need to travel something even shorter than that, etc."

The problem lies in the sort of assumption that adding up an infinite number of terms equals infinity. Some infinities are "larger" than others. There is technically no such thing as "the number infinity." Infinities only make sense when you speak of limits, and when we're talking about limits, we're really talking about rates.

Even with a fully continuous underlying space, Zeno's Paradox still fails. An infinitesimal amount of distance may be infinitely small, but it isn't nothing.

*****

I also want to point out to you guys: There's a ***HUGE*** difference between quantizing space and making space discrete. You don't need to invoke discrete space to solve this problem.

[Reach]

I also want to point out to you guys: There's a ***HUGE*** difference between quantizing space and making space discrete. You don't need to invoke discrete space to solve this problem.

But space is not continuous, which in and of itself does solve the cognitive aspect of the problem. It might not be necessary, but it's an easy solution.

This is, as far as I know, also a very common solution to this problem, so I fail to see why you have an objection to this.

Even with a fully continuous underlying space, Zeno's Paradox still fails. An infinitesimal amount of distance may be infinitely small, but it isn't nothing.

Everything you've said up until here makes sense, but this doesn't. It's a contradiction. What do you mean?

Your second statement is exactly why Zeno's Paradox is so hard to wrap your head around; it's not a solution. Because infinitely small distances are still distances, and because there are an infinite number of them if you treat this problem as a series, you arrive at the paradox.

You haven't provided a solution to the paradox, other than " In this case, the fact that you're talking about a finite distance already solves the problem.", which is fine and obviously right LOL, but it doesn't 'explain' anything.

(Yes, everybody and their dog knows the sequence can be defined in a finite manner using a convergent series. At least, if you've taken Cal. It's a terrible dissatisfying answer though, IMO, because it doesn't even answer the fundamental question the paradox is asking in the first place.)

[qqwref]

Originally Posted by Reincarnate

The problem lies in the sort of assumption that adding up an infinite number of terms equals infinity.

I wouldn't say this assumption is made in the paradox at all. In fact, it's implied (even if you know no math) that the sum of the infinite number of terms is finite. The thing that makes Zeno's Paradox confusing at all is the implication that, to move at all, you have to complete an infinite number of tasks - even though you might think you could complete them in a finite time, there are still too many to ever count.
of course, the Paradox is a load of bollocks anyway, but it does require some thinking/explanation the first time you see it
To me, the issue Zeno's Paradox brings up is actually quite distinct from real-world questions such as "does this ball bounce an infinite number of times".

[Reincarnate]

Reach: Whoa, whoa, whoa there.

"Space is not continuous" has *definitely* not been proven one way or the other. One of the problems with combining GR and QM is that QM states that space must be quantized. Applying these quantization methods to gravity fails on a GR scale. To say that something is quantized doesn't mean that there are discrete quanta of space -- it just means the objects and operators you discuss need to have quantization methods applied to them.

So when we talk about things like Planck lengths, it's an assumption made by the model that you arrive at through dimensional analysis. It doesn't say that things can't be smaller than Planck length -- it just says that beyond this point, we can no longer probe them. To go beyond the dimensional analysis, we'd need to invoke a new physical theory.

Like I mentioned, Zeno's Paradox is confusing to people because they think that adding up infinitely many pieces means that you're somehow arriving at an unachievable infinity even though we're talking about a finite distance. Yes, we could think of chopping up distance (or time!) into smaller and smaller units -- and we could think about doing this forever -- this doesn't mean the act of traversing distance or time needs to also take forever (which is how we incorrectly arrive at the confused conclusion that we should be unable to move and that time should stand still).


qqwref: It's the assumption that confuses people. They think "in order to get from point 1 to point 2, I must first travel to 1.5, but then to get there I must travel through 1.25, etc" and we could do this forever. So it confuses people into thinking "If we can chop up this distance into infinitely small pieces, how can we possibly get anywhere if we always have to go to an intermediary first?" This is what I mean by "adding up an infinite number of terms equals infinity" as the confusing implicit assumption (it's the same thing as when you say "The implication that to move at all you have to complete an infinite number of tasks").

[Izzy]

I'm still going with that the assumption of the paradox is flawed. It makes sense that once you get small enough there is no smaller unit of space which means that there is always a finite number of these units between 2 points.

Only conceptually can you continue to divide the measurement in half, but physically this will inevitably become impossible. But even if you could infinitely divide a measurement in half causing an infinite amount of points between 2 points I don't think that automatically means that everything is impossible due to some kind of limitation on time.