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A question I need answered.
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!? |
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 |
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.
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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 |
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.
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Re: A question I need answered.
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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. |
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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" |
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. |
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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. |
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. Quote:
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". |
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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. |
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. |
Re: A question I need answered.
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. |
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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. Quote:
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.) |
Re: A question I need answered.
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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". |
Re: A question I need answered.
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"). |
Re: A question I need answered.
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. |
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You just don't provide an answer to the fundamental cognitive problem the paradox provides, which is what qqwref is getting at. |
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can't help but think this is described by something like this:
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Re: A question I need answered.
Technically qq and I said the same thing in different words. The problem of Zeno's Paradox is the notion of traveling to an infinite number of places in a finite time.
But no, the Planck length issue means that we simply can't probe it. It doesn't mean that space is discrete like pixels or something. Even when we talk about pixels on a screen, we can still describe aspects *about* them using continuous mathematical concepts, and likewise for elements that involve sub-Planck-lengths. When we talk about Zeno's Paradox, we would basically be saying that any movement that occurs could only be observed down to the Planck scale for certain units of time. But this is still not dt time and dx distance we're talking about it. We're talking about something larger, and so it just means that we can't *observe* anything smaller even if they take place and take some sort of measurement. Think of it like using a camera with a finite FPS counting objects whizzing by that eventually move faster and faster. The camera can only probe so much. It doesn't say anything about the stuff it can't detect. And so assuming that spacetime is somehow discrete is a sort of handwaving solution that doesn't really address the reason this Paradox is tough: Calculus. To invoke discrete spacetime is to basically say "We'll remove the psychological problem we experience with this paradox by fixing the division of space with an eventual stopping point" when the real crux of the paradox is that the psychological problem need not exist in the first place. The approach you propose just fixes the symptom and misses what the calculus is actually saying. The implications of what Zeno is saying is that there is a halfway point between where you are and where you want to go... as long as you're not yet at the finish line. The problem with infinity is that if you're going to talk about finite concepts you have to imagine that you're at the "end" of an infinite process. Zeno is trying to get you to think of it in terms of something you count through, much like how you might count through writing .3 repeating. If you fall into this trap, you'll never make it. No matter how many 3's you write, you'll never be at 1/3. No matter how much you subdivide Zeno's distance, you are by definition subdividing the distance and are logically leaving pieces behind that you aren't dealing with in your future calculus of further divisions. But that's why the solution to the paradox "feels" tautological and not very satisfactory. It's all due to the nature of infinity. If we can travel distance x in time t, then we travel .5x in time .5t and distance dx in time dt. If we're talking about infinite divisions, then dx and dt are valid concepts to talk about. This means we ARE moving some distance in some time -- both nonzero. We can keep subdividing distance, but this means we're also subdividing time and we fall into this trap where we're "counting" things. But when we're talking about dx and dt, this also means we can *actually arrive* at the full-blown destination. That's the entire reason a limit works. We can keep subdividing, but if we subdivide to infinity, that means we can actually talk about MOVING that distance in that amount of time. In other words, yes, to arrive somewhere you must arrive at its midpoint first. Assuming we can move at all, we'll get there eventually. The subdivision argument is a psychological/logical trick that will ensure you never solve the paradox if you attack it from that angle. |
Re: A question I need answered.
As much as I didn't need it, thank you for the elaboration Rubix.
It was completely and utterly missing from your original blurb on the subject. As an aside, discussing space continuity further will only break this thread, so I will only add this: How are you defining continuous? A mathematical space is only defined as continuous if it's metric can undergo infinitesimal subdivision. When I talk about discontinuous space time, I'm talking about distance being undefined beyond a certain threshold, causing a breakdown of metric continuity. The plank length is exactly this scale. I wasn't aware that there was any dispute over this. It's what I was taught in quantum physics. |
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It's not so much a dispute as it is a notoriously tricky concept to grasp. It's hard to distinguish between quantizing something and making something discrete because they both sound like they're saying the same thing. We can talk about discrete energy states and technically be referring to "quantized" states at the same time. When we start talking about space itself, though, "discrete" loses its meaning because we're not talking about something countable like energy states.
We just say that the Planck length is the smallest unit where things make any physical sense. At lengths/times less than one Planck unit, quantum theory no longer applies to that realm. We don't have a good, solid quantum understanding of GR yet so we try to determine realms of relevancy by combining GR constants (G and c) with QM constant (h) to result in the fundamental units. So yes, Planck length is the smallest possible length we can talk about meaningfully, but really it's just that we don't have any reason to believe one way or the other than our current theories have any application below that scale. |
Re: A question I need answered.
How about this:
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f(x) = int[g(x) / p]x is time elasped g(x) is a function someone would normally use to get distance in meters, assuming continuity in time and space. p is the number of Planck lengths in a meter f(x) is the actual distance in meters, assuming that space is divided into discrete units. int is a function that truncates the decimal portion of any real number. I don't know if "int" is the appropriate function, as opposed to "ceiling" or "round", so feel free to discuss. I was just thinking about the topic and I came up with the above formula. |
Re: A question I need answered.
A note on all this stuff because it pisses me off when I see people treating Planck stuff this way:
A Planck length is 1.61605*10^(-35) m, so that means there are 6.18792735 * 10^34 Planck lengths in a meter. That's all you need to convert. You get Planck's constant from (hcross*G/c^3)^.5, where all the units cancel out to give meters. Planck time is 5.39124*10^(-44) seconds, which is derived from (hcross*G/c^5), which is the amount of time it takes the speed of light to travel one Planck length. You all know Planck speed, c -- an upper limit on speed, the speed of light. Now for the interesting part: Planck mass is (hcross*c/G)^.5, or about 22 micrograms, or the mass of a Planck particle which is a black hole with a Scharzschild radius (or event horizon) equal to Planck length. You'll notice it's actually quite large compared to everything else. This mass represents the smallest possible mass that can collapse into a black hole (which is good news for us, considering that most elementary particles are smaller than this). Note that GR has no such restriction on minimum sizes of black holes. All it predicts is that if a mass is squeezed smaller than its Scharzschild radius, it collapses into a black hole (the radius here is Gm/c^2). But on the quantum level, we talk about Compton length, where quantum effects become dominant (given by h/mc). These two figures are inversely proportional to each other, which is why we typically treat GR and QM as mutually exclusive theoretical frameworks. However, when you thrust these units together you find that QM and GR are both dominant at these levels where the Schwarzschild and Compton radii are equal. Planck density (specifically, Planck mass/(Planck length)^3 or c^5/(hcross*G^2)) gives you the largest meaningful density -- which also happens to be the exact same as the density of the universe one Planck time after the Big Bang (Planck temperature is the highest possible temperature -- a radiation with a Planck-length wavelength). These lengths, masses, and times are just constraints derived from the frameworks of the underlying theories. These units say nothing about what may or may not be actually happening at levels beyond these. At any rate, we haven't even begun to plunge into some of these levels yet. We're still multiple orders of magnitude away from a Planck second or Planck length. These units are limits on quantum field theory and classical gravity. Here you start getting into quantum gravity stuff (otherwise, trying to marry GR and QM by just hard-fusing shit from both sides at this level results in some LOLworthy mistakes where vacuum energy is over 100 magnitudes off). To give you an idea of the magnitudes, we all know that if you were to blow up an atom to the size of a football stadium, the nucleus might be the equivalent size of a pin placed in the center of that stadium. But say you blew that atom up to the size of the whole goddamned observable universe. A Planck length might be the equivalent of a typical Earth tree. Basically: "Substances are the smallest units! Grains of sand! Dust! We're right!" "No, no, atoms are the smallest, duh-doi. We're right!" "JK protons and electrons fo shizzle. We're right!" "WAIT! Stop the presses! QUARKS! We're right!" "Planck lengths, dude! We're right!" I think you see where this is going. All these limits are derived from the basis of the models in which they're derived. Planck mass alone should give you a pretty big clue. We derive it with the same sort of methods that we derive Planck time and Planck length, and yet we experimentally know that there are things larger and smaller than Planck mass. Planck mass is a limit *within the confines of the frameworks you're discussing*. Planck length and Planck time just happen to be smaller and faster than anything we can probe. But that says nothing about potentially new, future physical theories that speak about things even more extreme. Regarding problems like Zeno's Paradox, this is a problem of calculus. This assumes that you're capable of observing something with infinite precision. Invoking discrete space is just a handwaving tactic to get out of trying to understand why the paradox itself is so tricky to understand within the arena it was built on (calculus). Even if you do invoke discrete spacetime, you're not really solving the paradox. All you're saying is that you can't infinitely divide space and time and therefore the entire problem itself is bunk to begin with. Even if you take this route, you're not explaining how the reality is actually working. It's like people think of discrete spacetime as pixels where, at the lowest possible level, you have one Planck-length pixel "lighting up" every Planck-time. But then you have to answer "what does it mean to light up every Planck time?" We're still talking about something finite -- something continuous, but subdivided up into a discrete scale. All this is doing is solving the initial psychological boundary and not the problem of infinity itself. Sorry for the long rant but holy shit I get fumed when I see math and physics so blatantly pillaged. |
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That's how it's pronounced. You can also call it hbar or reduced h.
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Re: A question I need answered.
I'm pretty sure h-bar is at least 100 times more popular. Compare:
http://www.google.com/search?hl=en&q...=g-b4&aql=&oq= http://www.google.com/search?hl=en&q...&aqi=&aql=&oq= (Don't look at the number of results, look at the top results - "h cross" brings up a lot of fake ones so the number's off) |
Re: A question I need answered.
I prefer using h-cross -- sounds cooler (it's also the way Balakrishnan pronounces it, and that guy is an excellent teacher on par with Susskind. I've watched all their videos on YT. The internet, in addition to books, is where I learn my physics so idk what most people call things when it comes to shorthand)
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Re: A question I need answered.
Holly shit guys...
I'll be back in a long time with a response to all this. |
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just respond now, the problem's already solved so idk what more there is to add
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If you blow up an atom to the scope of the universe and a planck is an earth tree, you can blow up an atom on that tree to the size of the universe and repeat the process an infinite amount of times.
Infinity is one hell of a concept. |
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You *could*, but we wouldn't be able to know anything about it yet
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This thread made me think of ZPE and stuff. If there are an infinite amount of spaces between a space, that means ZPE isn't possible, right?
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There are an infinite number of spaces, yes, but the trick is that we actually pass through all of them
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:< fffffffff
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