Physics: Decimals can't be infinite because the space between must end. - Page 2

Just asking this again because the question was buried in a post above:

Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared) ?? I Just can't bloody remember how to calculate it.

[QUOTE]Originally posted by sunilraman

Well, all I want is to be able to go to my internet discussion forum for rumors about upcoming Apple Computer products without bumping into a bunch of nerds!

ROFL

In response to how would one calculate how many steps you would have to take to get 1 A separation:

log (10^10)/log(2) = 10*1/(log(2)) = 33.2192809 (looking up on a log table )

Whoa... Cool. Logarithms. Takes me back to high skool maths. Completely forgotten all that stuff. Don't do drugs, kids!! Now, hardeeharhar, can you answer the deceleration rate question?

If I had a spread sheet in front of me (Kaleidagraph really, since it can do derivatives)...

Ahhh. I see someone making assumptions about the motion!! (I thought scientists didn't do that in your world) Do we know if the motion is smooth, or is it really herky-jerky like the earlier mentioned kids who would take steps and pause??? You can only determine the part to the right of the decimal if you know the equations of motion hence my earlier answer that it is sometime between the 33rd and 34th second. Is it OK to be sloppy when the equations are "nice"? But all lingoistic when writing in prose?

Actually I already knew 2^33 is about 8 billion and 2^34 is about 16 billion. I used the spreadsheet to verify I hadn't screwed up my decimal places.

Three columns, one for n {looked at 28-42}, one for 2^(-n) and one for 1^(-10). Then did a quick visual comparison to double check the intervals where the values crossed.

It doesn't matter if the steps are jerky or nice and neat and decimilic. The formula still works.

To some extent this is the rationale behind quantum mechanics, which is a subject where I'm not even a casual expert, and I won't make much of a claim.

But I will say that, at the subatomic level, energy tends to operate in quantized amounts. This has been observed in many classic experiments of the 20th century.

The best part about math is that "imaginary numbers" are very much part of reality.

[QUOTE]Originally posted by sunilraman

Yeah, but a pragmatic approach is virtually pointless when tackling a mathematics problem. Sure the math problem is placed in the real world to make it more approachable, but it has no consequence whatsoever as to the actual explenation. Which is, of course, that an infinite addition of infinite small numbers does not get you an infinite large number. This may go against your language-based intuition, but should not go against you mathematical intuition.

The easiest way to 'prove' this, is trying to prove the opposite: that the number will grow to infinite. Well, let's try:

1 + .5 + .25 + .125 + ...
Now, everybody can see this number will never grow bigger than 10. Or 5. Or 2. But it will grow infinitely close to 2 (meaning you can prove it will grow bigger than any number smaller than 2, no matter how close to 2 it is), for all practical ánd mathematical matters (and here is where the two meet) making it equal to 2.

Edit: i never had to explain math in english so excuse me for any weird wording.

Gaaaaahhhh! Zeno's Paradox has nothing to do with quantum mechanics!

Here it is again, restated: Since an arrow flying towards a target must traverse half the distance, then half the distance remaining, and half again, and we can continue to halve the remaining distances forever, the arrow therefore can never reach its target.

This is not resolved by talking about quarks, or electron clouds, or the uncertainty principle, or anything in that neck of the woods. The arrow is not imagined to enter, in the last pico seconds before hitting the target, some kind of boundary ambiguity that resolves the paradox, because that is not remotely what the paradox is about-- unless somebody is prepared to argue that, in fact, the arrow never does reach the target and continues to move through an infinite series of bifurcations, forever.

Which it doesn't, and is in no way a condition suggested by quantum mechanics.

The paradox is simply, and again, about the difference between an infinite sequence tending toward limit x, where x is the distance to the target, and the real world movement of the arrow. The paradox hinges on the ambiguity of natural language being switched for mathematical description, not on some inherent ambiguity in the idea of "things touching".

What utter horeshit. Clearly, you have only been getting one side of the Zeno's Paradox debate, and that, no doubt, is coming from mainstream math, which has a vested interest in maintaining its hegemony. And they will brook no dissent, as your snarky look down your nose just indicated. Whether you like it or not, teh relationship between Zeno's paradox and quantum mechanics is a subject of debate, and sticking your fingers in your ears isn't going to make it go away. If I have some time, I'll dig up some of the papers I have on paleo-Zeno's Paradoxic change over time. You'll find, once you read up on the subject, that the "hockey stick" graph of the frequency of halfway points as the distance between objects decreases is actually a distortion and based on a limited number of experiments that have never been repeated.

You owe me a keyboard, sans snorted out coffee.

I'm glad I've given up maths in college. Although I am doing physics. Poo.

I'm sorry, but that's just bull. Zeno's Paradox was all about the 'mainstream math' problem as discribed by Addabox and myself, and has nothing to do with mechanics. Sure, today it is used to illustrate some theoretical concepts about things 'touching', but is completely unrelated. It's an interesting and worthwile topic, but does nothing to 'solve' or 'answer' Zeno's Paradox.

Sorry, SpcMs, Midwinter is just having some fun with the tone of another thread about global warming. He's just really good at imitating the style of an obtuse loon. (Hmmmm....... or is he?)

I would send you one, but it would never get there.

See? This is how it starts.


I see you've added a new one:

f) you don't seem to know what the hell you're talking about.

And so it has begun. The post-objective truth era begins to spread through the forums.

Soon, we can question whether the "Mac" actually exists at all, or if it is just a liberal hobby-horse propped up by a cabal of MS hating terror lovers.

Oh it's far worse than that, my friend. Postmodern mathematicians are destroying traditional liberal education and indoctrinating our children with their so-called "mathematical relativism." All you have to do is look at the debate over Fermat's last theorem to see it in its fullest flowering, where it's become a kind of Rube Goldberg-esque game to see how complicated the answer can be. Anybody with half a brain knows the solution is "equals 4."

We must be ever-vigilant about this if we are to preserve our way of life, because these people truly do hate America and all she stands for.

Two keyboards.

Not if there are any kinks in the derivative curves, or if accel is non-linear between sample points.

I find the debate on Zeno's utterances infinitely, even limitlessly more satisfying at the moment though.

When did I ever mention Zeno's Paradox? I was responding to the author of the thread, and happened to notice that none of the physics-based answers really touched upon the basic idea that energy is released in distinct packets.

Beyond that, if there's a universal speed limit, i.e. the speed of light, and it happens to be a quantifiable amount, then what prevents the quantization of minimum amount of energy? If I'm not playing along with your little game, I apologize, but I'm not interested in Zeno.

You're discovering the difference between mathematics and physics.
Mathematics is ONLY concerned with being consistent with itself.
Physics is concerned with explanations of physical phenomena that is consistent with our observations.

The distance between objects at time = t (sec) is 1/ (2^t) meters.
Acceleration is second derivative of distance function.
That's twice the deceleration of each object.
I'll let you ponder the simple derivative. :-)

The acceleration between points doesn't matter (remember the pathway independence concept in all physical sciences)...

But really whatever.

Sigh.

Please read CosmoNut's original post again. It's in two parts:

1) CosmoNut notes that he understands that the distance between two objects can be halved repeatedly, forever, yielding a number that constantly approaches, but never reaches, zero (or, to put it another way, the sum of these numbers approach, but never equal, the original distance).

2) He then wonders how that it is that objects may be observed to traverse this seemingly "infinite" space (the space getting smaller between two objects approaching each other part isn't necessary, the principle is the same for any finite interval).

This is Zeno's paradox. Sorry it doesn't float your boat, but that's what it is. It isn't a physics problem, and pointing that out isn't "my little game". Everybody is free to talk about anything they want, but CosmoNut's original question is very definitely "So, what's up with Zeno's paradox?"

[QUOTE]Originally posted by skatman
The distance between objects at time = t (sec) is 1/ (2^t) meters. Acceleration is second derivative of distance function.
That's twice the deceleration of each object.
I'll let you ponder the simple derivative. :-)


Sorry skatman, can't process this stuff. Why can't anyone tell me what the deceleration of the object in metres per second squared?

*All I remember is that if you have an acceleration curve, you differentiate at a certain point you get velocity.... or something like that.

[QUOTE]Originally posted by addabox
Two keyboards.

[QUOTE]Originally posted by midwinter
I would send you one, but it would never get there.


ROFLMAO

edit: Zeno's paradox is somehow related but Let's go back to CosmoNut's question and the title of this thread.

"Physics: Decimals can't be infinite because the space between must end."

[QUOTE]Originally posted by CosmoNut
I understand the principle behind the assertion that decimals never end and that you can (theoretically) infinitely halve the space between two objects and they'll never touch. A parabolic curve works on this principle, but really...

[QUOTE]Originally posted by CosmoNut
If two objects are travelling at each other, at some point the space between them must end so that there is NO space left and they are touching, right? By this notion, I'd think that a value of their distance apart MUST end. So it seems that at some point 0.00000000000000000...1 must become simply 0 when the objects finally touch.


You all are arguing in a totally different space of thought as to what CosmoNut is saying. In the first part he is talking about a series of infinitestimally small numbers, then jams on the idea to a real world physical idea.

Thus any discussion here should involve both physics and applied maths

In CosmoNut's second part, he confuses the issue further by saying that he makes an assumption 0.0000.....0001 must equal zero, therefore two objects extremely super-close to each other must therefore also have zero space between them.

Yes Zeno's paradox is related in some way, but it is a bit different in that it talks about an arrow flying to the air, at each stage (slice of time) it is not moving, therefore on the whole it never moves at all. The paradox is proven false in general by looking at it and saying, you can't just decide that time is discrete and slice it up the way you might do with an infinitely small series of numbers. Ie. 0.00000000....0000000001 sec is not equal to zero.


So we go back to try and explain in a way that deals with the *PHYSICS* aspect of CosmoNut's question, eg two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc. Will they ever touch? Well, around 33seconds, the distance between the two objects is about 1 Angstrom, one-ten-billionth of a metre. Quantum mechanics start to come into play then. From 33seconds onwards that's where it gets "all sub-atomic and shit" because the definition of "touching" then comes into play.

Similar in a way to disproving Zeno's paradox of the arrow is saying that just because you can split numbers into superbly small amounts doesn't mean that space or time in "the real world" works that way. Physics people are still exploring what the smallest amount of space is (they don't just say 0.000000....00000000001 metres must equal zero) because subatomic space just doesn't work that way. Similarly, they don't just cut up time into superbly small amounts and then say 0.0000000.....0001 seconds must be zero. If they did this, then it would create a situation where time did not exist at all. First and foremost this violates Newtonian physics. It also affects quantum mechanics because for example, the position of an electron in an atom is not static. It pops here and there (mostly within the electron orbital) OVER a certain amount, however small, of time.


So essentially, we go back to the point that in Maths you can do a lot in pure maths that doesn't have to relate to the real world. However, when you use certain aspects of Maths and try and apply it to the real world, eg. CosmoNut's question and Zeno's paradox, it comes straight up against the Physics view of the world. Whether you're coming from the Maths side of things or the Physics side of things the moral of the story is that when combining the two you can't just simply BASH TOGETHER one idea from Maths and another idea from Physics and hope it flies (pun unintended wrt Zeno's arrow)

[QUOTE]Originally posted by max_naylor
I'm glad I've given up maths in college. Although I am doing physics. Poo.


Can you answer my deceleration question please? Nobody else here feels like doing it: Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

What is the rate of deceleration of each object in metres/(second squared)?

Is "extremely super-close" some high-falutin' technical math/physics term?

[QUOTE]Originally posted by Hiro
Not if there are any kinks in the derivative curves, or if accel is non-linear between sample points.


In my question, I was asking how long it takes for the distance between the two objects to equal 1 angstrom. Two objects start moving at each other, 1/4 of the distance each second. Initial distance between two objects is 1 metre at 0sec. At 1sec, distance 0.5 metres, 2sec, distance is 0.25sec, etc.

For the sake of the problem, it was assumed that the pattern of movement as described above is constant. No kinks

[QUOTE]Originally posted by midwinter
Is "extremely super-close" some high-falutin' technical math/physics term?

Okay now you owe addabox two keyboards, and you owe ME one keyboard and some nasal- sinus- cleanser-spray.

Now you just exposed yourself as espousing bullcrap. How much else have you made up? Lingo can't save you from that gaffe.

The fatal counterexample: an acceleration step function. No movement until 33.5 sec (later than your log result), then constant velocity (instantaneous infinite accel) and immediate decel back to zero velocity. Your log function computes an incorrect time because it implicitly assumes a smooth non-kinked acceleration function. It can't take into account that movement may not even commence until after the smooth functions result, leading to an obvious contradiction.

The fact infinite accel is theoretically impossible is of little consequence since there are still an infinite number of other accelerations that can also be applied after 33.5 sec that will get our object to the correct point at 34 sec, but it only takes a single counterexample to prove your statement false.

[QUOTE]Originally posted by sunilraman

The acceleration changes over time.

The proof of this goes something like this (and it's easier to shof for the whole system rather than one side at a time):

If we assume constant acceleration the equations of motion are as such:

The total collective distance traveled is 1-2^(-t); 1/2 at t=1, 3/4 at t=2, etc

also distance travelled = Vo + 1/2At^2

making 1-2^(-t) = Vo + 1/2At^2

Vo happens to be zero as no motion at time zero so that term drops out leaving 1-2^(-t) = 1/2At^2

simplifying gets us

2 [ 1- 2^(-t) ] = A
t^2


Here we see A changes as t changes. A contradiction of the equations assumption that acceleration is constant.


I don't remember off the top off my head how to derive the integrated acceleration equation which would allow us to solve for an instantaneous acceleration and velocity. I'll leave that exercise to other brave souls. Maybe hhh wants to redeem himself.

This thread is seriously in need of the proper engineering terminology, which is:

"Close Enough".

Bwahahaha!! You started this thread, CosmoNut, and like you and I predicted, all the lovely math nerds came out of the woodwork. I love it.

I say to Hell with Math... I'm an art major.