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]

Re: A question I need answered.

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]

Re: A question I need answered.

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

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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.

[benguino]

Re: A question I need answered.

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]

Re: A question I need answered.

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.

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]

Re: A question I need answered.

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]

Re: A question I need answered.

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.