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        riddles >> hard >> Walking On A Stretching Rubber Band
        
(Message started by: william wu on Jan 17^th, 2003, 4:27am)

Title: Walking On A Stretching Rubber Band
Post by william wu on Jan 17^th, 2003, 4:27am

A taut rubber band connects a wall to the back side of a toy racecar 1 meter away. Starting at time t=0, the racecar drives away from the wall at 1 meter per second, stretching but never breaking the rubber band. At the same time, an ant resting on the rubber band near the wall starts to move toward the racecar at 1 centimeter per second. Does the ant ever reach the racecar? If so, in exactly how much time? Why?

Note 1: Treat the ant as a dot, like in geometry. Its initial position at t=0^- is 0 distance away from the wall.
Note 2: Assume that the angle of incidence between the rubber band and the wall never changes. (In other words, this is all happening along one dimension.)
Note 3: Does this remind you of something related to cosmology?

Title: Re: Walking On A Stretching Rubber Band
Post by BNC on Jan 17^th, 2003, 9:15am

It’s been a long time since I did formal math, but here goes:

I will use D for “delta” and d for d (differentiation)

I will mark the ant’s speed with va, and the car’s speed with vc.
Since the ant is moving with the rubber band as it is being stretched, the movement of the car helps the ant along.
The equation I’m getting is:

x(t+Dt) = x(t) + va*Dt + vc*x(t)/(1+t*vc)*Dt

The “va*Dt” is the ant self movement
The “vc*x(t)/(1+t*vc)*Dt” is the ant movement thanks to the car movement: 1+t*vc is the band’s length, x(t)/(1+t*vc) is the relative location of the ant.

Changing from D to d (reducing time span toward zero – what is the term for that?), we get:
x(t+dt) = x(t) + va*dt + vc*x(t)/(1+t*vc)*dt

dx/dt = va + vc*x(t)/(1+t*vc); x(0)=0

Solving that yields:
x(t) = ln(1+t*vc)/vc + va*ln(1+t*vc)*t

OK, vc=1, va=0.1, so now the equation to solve is:

ln(1+t) + 0.1*ln(1+t)*t = t+1 (t+1 is the band length)

Solving, we get:
t = 21935.3 sec

thus the ant will reach the end after about 6 hours.

Is that right? (probably not, but where is the error?!)

Title: Re: Walking On A Stretching Rubber Band
Post by aero guy on Jan 17^th, 2003, 12:14pm

Hmm, I think you have a problem, but my answer is a bit out there so we shall see.

I agree with you up to and including your dx/dt equation, but in your next equation you are missing a va in the numerator of your first term. The general form of this equation where c0 is the initial length of the rubber band is:

x=(vc*t+c0)*va/vc*ln((vc*t+co)/c0)

I checked it and am pretty sure about this. The second mistake was saying that va=0.1, 1 cm per second is 0.01. I subbed in x=c=vc*t+c0 to the above equation and got:

t=c0/vc*(e^(vc/va)-1)=e^100-1=2.68812*10^43 s

This is about 8.5181*10^35 years (including leap years)

The universe is estimated to be between 15 to 20 billion years old (depending on your source), so it would take the ant about 4.259*10^25 times the age of the universe.

Since an ant lives between 3 to 18 years depending on the species and whether it is a queen or drone or worker, it would take about 4.732*10^34 ant queen lifetimes to run this little race, so I think we can safely say that it will never make it, or if it is an immortal ant, that the universe will end before it gets there.

If, however, you meant 1 DECIMETER/second, then it would only take our little six legged friend about 316.88 years. So, even though he would long since have shuffled of this mortal coil, at least he would beat out the heat death of the universe; or the big crunch depending upon which theory you subsrcibe to.

Aside from the obvious cosmological level, or supercosmological level of numbers we are dealing with, it very vaguely reminds me of approaching the speed of light, but I think the analogy is a bit stretched.
I think it is very likely I missed something consdiering the answers I got, but I'll be damned if I can find it. I think it was enough of a stretch to have to look up integrating constans, yuck. Are these the numbers you were looking for, and what was your cosmological analogy?

Oh, it is interesting that for any non-zero speed of the ant there some answer, he will theoretically always make it.

Title: Re: Walking On A Stretching Rubber Band
Post by aero guy on Jan 17^th, 2003, 12:21pm

Oh, and I think this riddle should be moved to medium. It involves a bit of math, but not too much thought. Maybe you could exchange it with the impossibly friggin hard cryptic address which I am dying for an answer on.

Title: Re: Walking On A Stretching Rubber Band
Post by aero guy on Jan 17^th, 2003, 12:23pm

OK, I think I get the cosmology, motion within an expanding universe?

Title: Re: Walking On A Stretching Rubber Band
Post by SWF on Jan 17^th, 2003, 9:02pm

Aero Guy, I agree with you formula, but when you plugged in the values you did not include dividing by 100, so your time is too high. The correct time is a more reasonable 2.66e41 seconds. :-)

I think the toughest part of this problem is solving the differential equation given by BNC, but agree that this belongs in Medium because it is standard method. The differential equation can be written down almost immediately: velocity of the ant relative to starting point= velocity relative to the rubber band plus velocity of the rubber band where the ant is standing. One way to solve it is to multiply the equation by the integrating factor 1/(1+v_c).

http://tcw2.ppsw.rug.nl/~towr/PHP/formula.php?formula={v_a+\over+1%2Bv_ct}={1\over+1%2Bv_ct}{dx\over+dt}-{v_cx\over+(1%2Bv_ct)^2}= {d\over+dt}{x\over+1%2Bv_ct}

This allows the rest of the equation to be solved by integration.

Title: Re: Walking On A Stretching Rubber Band
Post by BNC on Jan 18^th, 2003, 3:22pm

Arrrg!
That’s what you get when you’re rusty. Lately, if I “did” any math at all, it would end at the differential equation stage. The rest would be done numerically on the PC. And here I go trying to solve using pen and paper.
Of course, you (aero guy and SWF) are both right. Sorry ‘bout that.

Oh, and I agree it belongs in the “medium” section – unless there is something we all missed…

Title: Re: Walking On A Stretching Rubber Band
Post by Icarus on Jan 18^th, 2003, 6:47pm

I saw this one originally in (where else?) a collection of Martin Gardner's columns. He had it as a bug crawling on a rubber rope. In addition to the ridiculously long time, one of his readers calculated that even if the rope were a meter in diameter to begin with, by the time the bug completed the trip, the distance between individual atoms in the rope would be greater than the diameter of the universe. Hope this ant is good at jumping! ;)

Title: Re: Walking On A Stretching Rubber Band
Post by william wu on Jan 19^th, 2003, 4:28am

on 01/18/03 at 15:22:24, BNC wrote:

Oh, and I agree it belongs in the “medium” section – unless there is something we all missed…

Heh, I knew someone would say this. No, it doesn't look like you guys missed something ... the answer is supposed to be very large, on the order of e¹⁰⁰. The analogy is indeed motion in an expanding universe. Ants on inflating balloons, all that stuff.

I think I'll leave the puzzle in hard. It's all subjective anyways. I struggled with it, and so did some friends of mine. Notably, I struggled with frames of reference, combined with much rustiness in continuum-related problems.

Title: Re: Walking On A Stretching Rubber Band
Post by James Fingas on Jan 20^th, 2003, 12:56pm

I think there is an easier way to think about this. I suggest that we keep thinking in terms of the original length of the rubber band. We're asking how long it takes the ant to crawl over 1 meter of rubber band, with his speed rapidly decreasing.

speed = 0.01/(1+t)

Since the speed decreases as a harmonic function, then no matter how slow the ant initially is, nor how long the rubber band is, he will eventually reach the end (because the integral to infinity of the harmonic function is infinite). Integrating the speed over time gives us the formula for distance (this is the same as other people have given):

distance = 0.01*ln(t-1)

However, I would argue that this doesn't really apply to cosmology because when you traverse the universe without accelerating, your speed remains constant with respect to your starting point. No matter how far you travel, you will still look ahead and see red-shifted stars. These stars are going away from you--you'll never reach them. In this puzzle, on the other hand, the ant keeps a fixed pace relative to the neighbourhood of the rubber band he's in. To do this with a spaceship, you'd have to keep accelerating (would it be at a constant rate?). The stars would keep getting farther and farther apart, but eventually you'd get to the end, because you'd approach the speed of light (as measured from your starting point). But that would take infinite fuel ... better hope there are gas stations along the way ;)

Title: Re: Walking On A Stretching Rubber Band
Post by aero guy on Jan 22^nd, 2003, 11:52am

SWF, dang I wish I could put the equations in that nifty format, but well, I guess I am too lazy, heck I haven't even registered. The integrating factor was the key thing, and I had to dig up a text book from a class I took seven years ago to find it (thankfully it is on my desk, basic calculus books often come in handy). I don't get the "divide by one hundred" part though. Are you referring to vc? It is one.

Title: Re: Walking On A Stretching Rubber Band
Post by william wu on Jan 22^nd, 2003, 3:02pm

on 01/20/03 at 12:56:02, James Fingas wrote:

However, I would argue that this doesn't really apply to cosmology because when you traverse the universe without accelerating, your speed remains constant with respect to your starting point. No matter how far you travel, you will still look ahead and see red-shifted stars. These stars are going away from you--you'll never reach them.

Hmm. Rather than comparing the racecar to a spaceship, how about a light pulse? Then the problem becomes whether this pulse will ever reach some far off location in the universe. I remember that an amazing and counterintuitive fact about light is that its speed remains constant regardless of the motion of the observer. (What was the proof of this?) So now that we've got our velocity fixed over all frames of reference, does the analogy work out?

Disclaimer: it's been a long time since I took physics, so what I suggested here could very well be rubbish

Title: Re: Walking On A Stretching Rubber Band
Post by ragna on Jan 22^nd, 2003, 5:32pm

I believe constancy of the velocity of light is actually a postulate of special relativity; if I recall correctly it had something to do with a thought experiment about chasing a light beam.

Title: Re: Walking On A Stretching Rubber Band
Post by SWF on Jan 22^nd, 2003, 7:34pm

Sorry about that aero guy, I see that your calculation is correct. I had typed in vc=100, but it is 1 like you say. I had double checked, but evidently that was not enough.

I don't think the analogy of this solution with light in an expanding universe is correct. The solution above used the assumption that velocity relative to the rope linearly superimposes with the boost due to stretching. Light will always travel the same velocity relative to the starting point, so it won't get an apparent boost from the universe expanding.

There is experimental evidence that the speed of light is constant, for example, the Michelson-Morley experiment.

Title: Re: Walking On A Stretching Rubber Band
Post by BNC on Jan 23^rd, 2003, 12:04am

The Michelson-Morley experiment is, according to urban legends, one of the triggers for Einstein's leap of genius. However, the constancy of c is a postulate. The results of the seemingly simple assumption are quite extraordinary. The results had been shown empirically to stand, but that does not "prove" the relativity theory, much the same as Newton's laws were never proved (until proved wrong).

One of the fundamental thought experiment that differ the Newtonian postulation (c changes) from the Einstein postulation (constant c) is this. Take a bucket full of water, and spin it fast around the axis. Naturally, the water surface will curve. Now the interesting part. Leave the bucket still, but spin the whole universe around it. Would the water surface curve? Einstein said it would.

Title: Re: Walking On A Stretching Rubber Band
Post by Icarus on Jan 23^rd, 2003, 6:34pm

So many comments...

Einstein did not learn of the Michelson-Morley experiment until after proposing the special theory of relativity. I believe he was inspired by the observation that Maxwell's equations, which had proven to fit the observed behavior of electromagnetism very well, did not obey the same rules of Galilean relativity (speed is relative to the observer) that the laws of mechanics did. Instead, Maxwell's equations predicted a specific speed for light. Others saw this and assumed that there must be something that this speed is with respect to, an "ether" which was responsible for electricity and magnetism. Einstein decided to examine the possibility that there could be a speed which was the same for all observers. It was this ability to question the assumptions that everyone else did not even know they were making, that was the real genious of Einstein (he did it again with his discovery of quantum mechanics).

I agree with BNC, that the special theory of relativity has not been proved, just like Newtonian physics has not been proved, and just like the theory that the sun will rise tomorrow has not been proved. Proof - to demonstrate that something must be true - has no part in science, and I cringe whenever I hear someone refer to "scientific proof" or a "science fact", as almost always the speaker misunderstands the nature of science (including practicing scientists).

Science is learning by induction. It is observing patterns, and extrapolating those patterns into the future. The problem is, just because a pattern holds in one situation, does not mean it will continue to hold in others. The good scientist knows this, that just because a theory has been successful in explaining everything it deals with so far does not mean that that something unaccounted for in the theory might change, and it no longer works. History is replete with examples of once-successful theories going to the dustbin, as further observations showed their shortcomings. (Ptolemy's universe did a very good job of predicting the behavior of the planets until a certain gold-nosed astronomer had to get meticulous in his measurements!) A good scientist holds no theory as sacrosant, and is always looking to test them in new situations. A theory that has successfully predicted a wide variety of outcomes without any failures is considered to be strong, but not proven. (Sorry to go on like this, but this is a hot-button issue for me.)

By this measure, special relativity is a very strong theory. In addition to the Michelson-Morley experiment, many experiments have been performed to test the predictions of special relativity. In particular, when radioactive particles are accelerated near to the speed of light, their half-lives increase exactly (to the limits of measurement) as special relativity predicts. This prediction has been tested thousands, or even millions (depending on how you count them), of times.

General relativity, on the other hand, is only lightly supported.

The statement about the water in a rotating bucket has nothing to do with the constancy of the speed of light or special relativity, or even general relativity for that matter. Newton would likely of said the same thing, pointing out that the two situations are not just equivalent, but the same.

Title: Re: Walking On A Stretching Rubber Band
Post by BNC on Jan 24^th, 2003, 12:27am

on 01/23/03 at 18:34:42, Icarus wrote:

So many comments...

Newton would likely of said the same thing, pointing out that the two situations are not just equivalent, but the same.

It's been many years since my first physics course, but I remember the lecturer then pointing that according to Newton, the water would not curve, as it can be shown that the Newtonian universe has (somewhere) a constant point of reference. Since the bucket did not move relevant to that point, it would feel nothing, and remain static. I may have to dig into my old notes to find the exact reference -- if I'll find the time (which I doubt).

Title: Re: Walking On A Stretching Rubber Band
Post by wowbagger on Jan 24^th, 2003, 6:03am

on 01/23/03 at 18:34:42, Icarus wrote:

I believe he [Einstein] was inspired by the observation that Maxwell's equations, which had proven to fit the observed behavior of electromagnetism very well, did not obey the same rules of Galilean relativity (speed is relative to the observer) that the laws of mechanics did.

Actually, it can be shown quite easily that the postulate of the same speed of light for all observers in inertial frames of reference (IFR) leads to the Lorentz transformation (describing change of IFR). This is the very transformation under which the Maxwell equations remain invariant, whereas they change under a Galilei transformation. As far as I know, this was already known before Einstein's work. Einstein's extended principle of relativity requires all laws of physics to be the same for every inertial observer, including the speed of light in a vacuum.

Quote:

Einstein decided to examine the possibility that there could be a speed which was the same for all observers. It was this ability to question the assumptions that everyone else did not even know they were making, that was the real genious of Einstein (he did it again with his discovery of quantum mechanics).

I couldn't agree more about Einstein's genius, but I wouldn't say "his discovery of quantum mechanics". People like Heisenberg, Schroedinger, Dirac and others were more involved in that (imho). Which isn't meant to depreciate Einstein's Nobel prize-winnning work on the photoelectric effect.

Quote:

Science is learning by induction. It is observing patterns, and extrapolating those patterns into the future. The problem is, just because a pattern holds in one situation, does not mean it will continue to hold in others. The good scientist knows this, that just because a theory has been successful in explaining everything it deals with so far does not mean that that something unaccounted for in the theory might change, and it no longer works. History is replete with examples of once-successful theories going to the dustbin, as further observations showed their shortcomings. (Ptolemy's universe did a very good job of predicting the behavior of the planets until a certain gold-nosed astronomer had to get meticulous in his measurements!) A good scientist holds no theory as sacrosant, and is always looking to test them in new situations. A theory that has successfully predicted a wide variety of outcomes without any failures is considered to be strong, but not proven.

Very well said! 8)
Let me add that the more verifiable predictions a theory is able to make, the "better" it is in a sense - until it's disproved.

Quote:

(Sorry to go on like this, but this is a hot-button issue for me.)

And it is a very important one as many ordinary people seem to have serious misconceptions about the scientific method or maybe science in general.

Quote:

By this measure, special relativity is a very strong theory. In addition to the Michelson-Morley experiment, many experiments have been performed to test the predictions of special relativity. In particular, when radioactive particles are accelerated near to the speed of light, their half-lives increase exactly (to the limits of measurement) as special relativity predicts. This prediction has been tested thousands, or even millions (depending on how you count them), of times.

General relativity, on the other hand, is only lightly supported.

Somewhere I learned that quantum electrodynamics is supposedly the best-verified theory (in physics) - regarding accuracy I suppose.

Title: Re: Walking On A Stretching Rubber Band
Post by Icarus on Jan 24^th, 2003, 5:51pm

on 01/24/03 at 06:03:25, wowbagger wrote:

Lorentz came up with his transformations before Einstein, though Einstein was not aware of his work and rediscovered them. But Lorentz's theory was that physical matter was affected by its travel through the ether, and that the contraction predicted was a physical squishing of the material. (How he explained the time contraction I don't know. Maybe that is why his theory was rejected). He did not view it as a change in IFRs. The failure of Maxwell's equations to conform to Galilean relativity was known, but as I said, the explanation everyone believed was that Maxwell's equations were only valid with respect to the mysterious "ether" that was responsible for electromagnetism. Thus if one is moving with respect to the ether, Maxwell's equations would no longer be valid. Michelson-Morley was an attempt to find out how fast the Earth was travelling with respect to the ether (since it orbits the sun, it must be moving through the ether). But to everyone's surprise, no such movement was detected. Lorentz's work was one of many attempts to explain this. It was only Einstein who considered the possibility of a true speed constant to all observers. An idea that violated concepts of time and space that very few people had even realized were assumptions.

Quote:

Einstein was responsible for very little of the development of quantum mechanics, not even of the core of it. That was done by the three you listed and many others (Planck, Bohr, De Broglie, Feynman, to list a few more). However, the very first idea of it, the first rejection of conventional wisdom, of assumptions so deep that no one recognized that they even existed, once again this was done by Albert Einstein. That is why I call him the discoverer of quantum mechanics. That was the absolute genius of the man. The rest are all great explorers of the land, but he was the one to first lay eyes on it!

Quote:

Somewhere I learned that quantum electrodynamics is supposedly the best-verified theory (in physics) - regarding accuracy I suppose.

I would think it would be hard pressed to beat the verification of Newtonian mechanics. Since the vast majority of modern machinery was designed by the use of Newtonian mechanics, the very fact that all these machines work as designed can be viewed as verification of those mechanics. quantum electrodynamics has been used to design very little by comparison. (I don't believe that QED is required in the design of electronics. Only simple quantum mechanics comes up. But I never got that deep into the subject so I could easily be wrong and the computers we are using would be verification of QED.)

A different objection I have to the statement though is that, just like Newtonian mechanics, QED is known to be NOT completely accurate. Instead it is an excellent approximation to the truth, so long as you don't look too close. The biggest problem is that QED and General Relativity are incompatible. This is why String theories and all their relatives came into being. ;)

on 01/24/03 at 00:27:40, BNC wrote:

What your lecturer was refering to was not a "constant point of reference", which definitely does NOT exist in the Newtonian universe. Such a point would violate the symmetry with respect to translation in Newtonian mechanics, which is in turn responsible for the Law of Conservation of Momentum. Since this law is strongly verified (and not even quantum mechanics breaks it), we can be sure such thing does not exist.

What he or she was talking about was the existance of "Inertial Frames of Reference". These are reference frames for which the laws of physics as we normally think of them actually hold. (In other frames of reference you have to add correction terms. For instance in a rotating frame, there is an extra term in the laws of motion indicating a radial acceleration.)

If your universe were empty except for the bucket and water, then you can assume that the universe is an inertial frame. In which case the water ought to curve to match the radius of the spin. If you assume that the bucket is an inertial frame, and the universe is spinning, then the water should stay flat. There is no contradiction in this as you can accurately say that these are different situations. The puzzle comes in, in that in your otherwise empty universe, who is to say when it is or is not spinning, or when the bucket is spinning in it? Since there is no reference point in this universe, there is nothing to compare the position of the bucket to, so as to determine the relative motion. Mach examined this question before Einstein came around, and came to this conclusion: it is the matter in the universe, through its gravitational interactions, that determines which reference frames are inertial. Since in that empty universe the only mass is the bucket and water, its reference frame is inertial, and so it can never be said to be spinning (other than around its own CG).

Now consider the bucket in this universe. We can try to take the view that the bucket is still and the universe is spinning around it. But then we have to consider the net gravitational effect of all that mass swinging around our bucket. Mach argued (whether he showed it in the math I do not know) that the resultant effect would be to cause the water to curve in exactly the same fashion as predicted by Newtonian mechanics for a spinning bucket in a still universe. (This is why I said Newton would have predicted the same thing as Einstein.) Thus it is the interaction with gravity that produces inertia. (This also explains why "gravitational charge" is the same as inertial mass: because inertia is a byproduct of gravity.)

All this holds in Special Relativity just as well as in Newtonian mechanics, as special relativity also has inertial and non-inertial reference frames. Hence any result relating to it has nothing to do with the constancy of c.

Einstein's involvement with this scenario has to do with General Relativity, not Special Relativity. I can't explain everything here. (I could go on all day and barely scratch what I know of the subject, which is just a bare pittance of what is out there.) But, part of the idea behind general relativity was to rewrite the laws of physics in such a way as to remove the special place of Inertial Frames, so that the same descriptions applied in every frame of reference. This is what his equation(s) of general relativity accomplishes. But he does not predict anything different about the spinning bucket/universe than Newton and Mach do.

This leads to my favorite result from Einstein:
Aristotle and Ptolemy gave us the Terracentric point-of-view: everything revolved around the Earth.
Copernicus gave us the Heliocentric point-of-view: everything revolved around the Sun.
This was followed by the Galactocentric point-of-view and the Acentric point-of-view.
But thanks to Einstein, all frames of reference are equal, and so I can claim the EGOCENTRIC point-of-view:
EVERYTHING REVOLVES AROUND ME! 8)