# Something fun for a change…Ant on a rubber band.

Does someone with an open mind want to try to tackle this? Its about a theoretical ant on a rubber string. If the ant is crawling on the string, but the string is expanding, can the ant ever reach the end. In the video they claim it can, because the ant’s progress gets stretched towards the end, at the same time the string is stretched. But this is a bizarre interpretation in my opinion. This is only true if the string is being stretched in only one direction, that is, the direction that it is approaching. So the string is secured at one end, and streched at the other end, so indeed it keeps getting closer to the end.

But if neither end is secured, and it is stretched, there is no reason for the ant to be stretched in the direction of its progress, thus it would never go anywhere. But neither the narrator, nor anyone in the comments seems to have a problem with this. Why?

And is this in any way analagous to spacetime stretching?

## 196 thoughts on “Something fun for a change…Ant on a rubber band.”

1. Phoodoo:

“And is this in any way analagous to spacetime stretching?”.

No. If it were analogous, the ant would be stretched at the same rate as the rubber band was.

2. One way to look at it: for any given speed that the ends are stretched, there are arbitrarily small subdivisions of the whole that will move more slowly than the ant. Therefore, since the ant can cross these subdivisions, and can count them off as it goes, it must eventually run out of them and reach the end.

Example: suppose you stretch the ends of an initially 1m band at 1m/second. The ant moves at 1cm/second. If you mark off the band at 5mm intervals, the marks will move apart at 5mm a second – half the speed of the ant, so it is continually crossing the marks. You don’t add any new marks, so eventually it runs out.

But what happens if you accelerate the stretching? In the example above, if you stretch at 1m/second/second, the marks outpace the ant after a couple of seconds. Is there an arbitrarily small subdivision in which the above reasoning holds? I’m not sure. It certainly seems intuitive that the acceleration will always outpace the ant eventually, however arbitrarily small you make the marks. But then again, there will always be a local region that is expanding more slowly than it is moving, so it must be going somewhere. Of course we soon hit relativity and quantum distances, respectively, which change the game again.

3. Allan Miller: But what happens if you accelerate the stretching?

Then the ant might not reach the end.
With a linear stretching rate, the percentage of the rope covered by the ant increases with log(time), so the ant will always get to the end, eventually, whatever the parameters.
Accelerate that stretching, and percentage of the rope covered is now, say, log(time) / time.

4. timothya:
Phoodoo:

“And is this in any way analagous to spacetime stretching?”.

No. If it were analogous, the ant would be stretched at the same rate as the rubber band was.

Well, I was actually refering to the stretching of the universe. As the universe expands, things are not stretched locally.

5. I am not sure what you are trying to say, but did you actually watch the video? It doesn’t seem you are refering to that situation at all. The reason the ant is making progress and not being pulled backwards is simply because of the direction of the stretching. If the ant was facing the other way on the string, and he still stretched the string the same, the ant would not be gaining with the stretch, it would be moving the opposite its direction walking. It seems clear that is why he attached one end when he did the demonstration.

I am not sure that you have addressed this issue that the video shows.

6. DNA_Jock: With a linear stretching rate, the percentage of the rope covered by the ant increases with log(time)

I am still not convinced by this. If the ant has covered 10% of a 50 meter rope, and the rope gets stretched to 100 meters, the ant has now only covered 5% of 100 meters. At least if you did the demonstration the way he showed and turned the ant around.

7. Nope.
In one sense this is analogous to the universe expansion question: phoodoo fails to comprehend it.
I am enjoying the way that phoodoo is claiming that it matters which end of the rope is held in place, and which end is being pulled. The fact that he claims this is all you need to know regarding his level of comprehension.
So to answer phoodoo’s original query “But neither the narrator, nor anyone in the comments seems to have a problem with this. Why?” because they understand basic geometry.

phoodoo: If the ant has covered 10% of a 50 meter rope, and the rope gets stretched to 100 meters, the ant has now only covered 5% of 100 meters.

No. Your ant has now covered 10% of a 100 meter rope. Take a rubber band, use a marker pen to mark it with a dot, and then stretch the rubber band (and unstretch it); if the dot is 25% of the way along the band, it remains 25% of the way along the band, however much you stretch it!

Here’s one way of looking at the math, phoodoo:
Let’s think about the percentage of the string that the ant has covered, P.
At the start, P=0. The ant will be successful when P=1.
Now, here’s the thing you are missing. If the ant clings onto the rope, then P does not change at all (think about the marker pen dot)
Let’s suppose that he moves down the rope at speed A (m/s)
Let’s say the rope has initial length L and is stretched by S (m/s).
Let’s use Pt to denote how far along the rope he is at time t
How far along the rope will he be a moment Δt later?
He’s moved A . Δt along a rope of length (L + S.t). Thus
Pt+Δt = Pt + (A . Δt )/(L + S.t)
Thus
δP/δt = A / (L + S.t)
Integrate:
P = A. ln(L+S t) /S +c
Whatever the values of the constants A, S and L, this function is always increasing and will eventually reach one.
Your mistake is thinking that the ant is somehow tied to his GPS co-ordinates such that, when the rope is stretched, it moves underneath him somehow. Hence your unique concern re which end of the rope gets pulled…

8. I was going to turn over a new leaf and not be insulting here anymore, but I think I might still make an exception in your case. Because the “math” guy still has no concept of reality. Go try your little experiement, put a “dot” on a rubber band, right in the middle and stretch it. What happens, where is the dot? In the middle you might say. But guess what, since your so exquiste math mind can’t actually use your eyes and brain, you are wrong. There is no “dot” anymore. It goes from a dot, to an oval, to a long smeared messy line. So you might call part of the once was a dot in the middle. But since it is not a dot anymore, and it is now a 20 inch long streak, what part of the dot are you talking about? Is this what happens to the ant? Does it get stretched from a half inch ant into a three foot long ant? And then a two mile long ant? And then eventually into a light year across long ant?

So it does matter where you strech, what is getting stretched and what is getting expanded. Is the space under the ant getting expanded, and is it getting expanded the same as one end, or both ends, or what. This is why I asked if this is analagous to what we imagine happening to the universe. Is everything being stretched? If not, the direction of stretch matters. Its not quite as simple as the depths of your imagination. Go get your rubber band, try it, and you will maybe increase your brain by 50%. That still won’t be enough, but….

9. phoodoo: Go try your dot on a rubber band and get back to me.

You mean like how you engaged with the mathematics of it as carefully laid out?

Moron.

10. phoodoo: Because the “math” guy still has no concept of reality.

You believe that Uri Geller can bend spoons with his mind. And that dogs can use PSI to tell when their owners are about to come home.

Tell us more about other people’s erroneous conception of reality why don’t ya.

11. phoodoo: .Go try your little experiement, put a “dot” on a rubber band, right in the middle and stretch it.What happens, where is the dot? In the middle you might say.

Instead of a stretchable dot, try using a paper sticker that can’t stretch. Try your experiment again. This will more closely match the case of the ant, which also can’t stretch.

Returning to the dot, I think there is some validity to what you write, because some models I’ve seen, extrapolating into the far future, do indeed see the contents of the universe stretching. In the somewhat nearer future, it’s predicted that distant galaxies will begin to disappear as the universe expansion rate exceeds lightspeed. Extrapolate far enough, and even atoms will be stretched.

Back in the present, stretching the dot makes no real difference. Think: the initial dot had width, which extended some percentage of the whole band. As the dot stretches with the band, that percentage does not change. And it STILL doesn’t matter where you stretch, so long as the rubber band is equally stretchable along its whole length.

12. If you put a sticker on a rubber band and then try to stretch it, one of several things can happen. The sticker can be strong enough and hold that part of the band, not allowing that space to stretch, while the rest stretches. This certainly can’t be the case in this model. Secondly, the adhesion to the band could be weak enough that the band slides out from underneath the sticker. In this case the ant stays where it is, in other words, its as if the ant is suspended above the rope, and not actually on the rope. The third is the sticker stretches or breaks. So is the ant suspended above the band, or does it stretch?

Because saying that the ground underneath the ant is the only part that doesn’t stretch makes no sense.

13. By the way, the percent of the whole rope, that Doc has tried to introduce is a red herring. Its his math brain that doesn’t understand that math has to have a correlation to the proper things that he doesn’t understand. Increasing the amount of the rope that the ant has already traversed does nothing for what it has left. This is his logical mistake. If the amount he has left keeps increasing, it doesn’t matter what happens to the amount already passed. It can grow for infinity and he still won’t get to the other end.

14. phoodoo: Its his math brain that doesn’t understand that math has to have a correlation to the proper things that he doesn’t understand.

A common lament from my former colleagues at theoretical biology is how they were always told that, because they had to make simplifying assumptions, their models did not correctly capture real world events. This was inevitably followed by a verbal explanation that grossly oversimplified things and tossed reality right out of the window.

I am sorry, phoodoo, but there is a flaw in your verbal model: even if the length of band the ant has to cover keeps increasing the ant will reach the other end. That’s because, as the ant is traveling, the absolute amount that gets added to the path ahead each time is also shrinking towards zero.

15. phoodoo: Its his math brain that doesn’t understand that math has to have a correlation to the proper things that he doesn’t understand.

phoodoo is seemingly unaware that the math can take into account all his objections. phoodoo prefers to deal with not-math. This also explains why phoodoo is unable to understand the concept of ‘fitness’.

16. Corneel: the absolute amount that gets added to the path ahead each time is also shrinking towards zero.

No, not if the strecthing is more than the ant is covering. So what is the ant doing, stretching, or hovering above the rope as the rope moves out from under him?

I understand why people just swallow what they are being told without thinking. Its the same idiocy that gives you ideas like some infinites are bigger than others, and Hilberts hotel can always find more space by having guests move over one room. Its dumb things like this that should tell you not to trust math symbols when they mean nothing. If you don’t think everything makes sense.

17. phoodoo: No, not if the strecthing is more than the ant is covering.

Yes, also when the stretching of the band exceeds what the ant is covering. As the ant approaches the far side, it will be increasingly less bothered by the stretching of the band.

phoodoo: So what is the ant doing, stretching, or hovering above the rope as the rope moves out from under him?

I assumed it is the band that is stretching, not the ant. Mathematically speaking, the ant is represented by a point that can move freely along the length of the band, adjusting its little legs to avoid doing a split.

phoodoo: I understand why people just swallow what they are being told without thinking. Its the same idiocy that gives you ideas like some infinites are bigger than others, and Hilberts hotel can always find more space by having guests move over one room. Its dumb things like this that should tell you not to trust math symbols when they mean nothing. If you don’t think everything makes sense.

Too bad you already gave up on your resolution to not be insulting here anymore. Personally, I like math a lot because it forces you to be explicit about your assumptions. Most people are oblivious of the implicit assumptions that go into their thinking.

18. Well, first, nothing I said was insulting or directed at you. I was speaking of people who believe math nonsense without thinking about it.

Now, about the ant. If the band is stretching, and the ant is not, then he can not be “carried along” in one direction of stretch. After all, if he is being carried by the stretching, why is he being carried in the direction of his goal, and not being carried in the opposite direction? There is the issue. If you imagine a dot on a rubber band, and you stretch the band, part of the dot heads in the negative direction, and part of the dot heads in the positive direction. Which way does the ant go when the band is stretched? Both ways? Neither? It doesn’t matter?

This is why I asked what is being stretched, and likewise when we talk of the universe being stretched, are you being stretched? Are you moving in one direction, or all directions, or no directions? Are you moving farther away from everything, so that you are not getting closer to anything? Analogies become hard to reconcile.

19. phoodoo: Now, about the ant. If the band is stretching, and the ant is not, then he can not be “carried along” in one direction of stretch.

Agreed, but this is not what people have been telling you is the solution to the paradox.

phoodoo: This is why I asked what is being stretched, and likewise when we talk of the universe being stretched, are you being stretched?

I do not know. I assume I am not, since objects have been drifting apart since the big bang. This is not possible if objects stretch along. I am guessing gravitational attraction keeps objects together.

20. Corneel: Agreed, but this is not what people have been telling you is the solution to the paradox.

This is what was demonstated in the video. As the band stretched towards the right, the ant moved in that direction. But why? Why doesn’t the ant move to the left, even though it is walking right? As we see with a dot on a rubber band, if it is in the middle, part of it can be stretched to the right and part to the left.

Couldn’t we just face the ant the other way?

21. phoodoo: After all, if he is being carried by the stretching, why is he being carried in the direction of his goal, and not being carried in the opposite direction? There is the issue.

This is fun.
He is not being carried in either direction.
In the video, Billy moves at 5 cm/s, and the stretch is 10 cm/s
W——-a——-b——-E
Suppose we place Billy at a point (a) one third of the way along the rope.
Looking eastwards, Billy sees the end of the rope receding at 6.66 cm/s.
Looking westwards, Billy sees the end of the rope receding at 3.33 cm/s.
Irrespective of how long the rope currently is (this matters).
[Note that it does not matter whether you think the East end is stationary or the West end is stationary, or if the entire caboodle is flying eastwards at 460,000 cm/s.]
Can we agree that, if Billy sets off promptly* to the west, he can reach the end of the rope?
Can we also agree that, if Billy sets off promptly* to the east, he can reach point b (halfway from his current position to the East end) that is receding at 3.33 cm/s?
If not, why?

22. Is Billy hovering over the rope? Is he being moved by the stretching of the rope? Is his body being stretched? Does this require math to answer?

23. Start it at time zero. W, A, B, and E are all at point zero. What speed does E begin moving away from A?

24. Here is the really fun part, for those who are playing along.

Watch the video beginning at 2:05. Then let’s go back and look at the clever mathematician Jock, who only knows numbers but doesn’t understand what words and concepts mean. Allow me to quote from the video:

Billy is no longer at 5 cm. BECAUSE THE KEY TO ALL OF THIS IS THAT THE ANT IS ON THE ROPE AND HE STRETCHES WITH THE ROPE!

Still gloating Jock? Still don’t think it matters which direction the rope gets stretched? Still think you are a genius?

25. Yup, this is the same problem with the concept of a point and the ability to correctly convert co-ordinate systems that we encountered a couple of times previously.

phoodoo: Is Billy hovering over the rope? Is he being moved by the stretching of the rope? Is his body being stretched? Does this require math to answer?

Billy is on a tiny pogo stick; depends on your frame of reference; no; and no.
BUT the answer to when Billy gets to the end does NOT depend on your frame of reference.

phoodoo: Start it at time zero. W, A, B, and E are all at point zero. What speed does E begin moving away from A?

The puzzle, as stated, starts with a 20cm rope, but I’ll play along. When W, a, b, and E are all at point “zero”, E is moving away from W at 10 cm/s, and E is moving away from A at 6.666 cm/s.
W-a, a-b, b-E are each moving apart at 3.333 cm/sec. These speeds never change.
[I think, in addition to the problems noted above, phoodoo is envisioning an exponential stretching (well, super-linear, anyway).]

phoodoo: Still gloating Jock? Still don’t think it matters which direction the rope gets stretched?

As you noted in the OP, none of the commenters seem to be worried about this either. It’s phoodoo vs the rest of the world here…
Now, I answered your questions, let’s see if there is anything that we can agree on:

I asked: Can we agree that, if Billy sets off promptly* to the west, he can reach the end of the rope?
Can we also agree that, if Billy sets off promptly* to the east, he can reach point b (halfway from his current position to the East end) that is receding at 3.33 cm/s?
If not, why?

26. Go watch the video again. Did you not understand the quote? Was the narrator lying?
I understand you have this paranoia about ever admitting you are wrong about anything. But I just quoted you THE EXACT reason you are wrong. He says the exact words in the video.

And yet Jock…

Well, anyone who wants can watch the video and see and hear for themselves.

Jock: “He is not being carried along…”

Video: “The key to all of this is that the ant is on the rope and he stretches with the rope.”

Do you understand what he means when he says “key”? Can it be any more clear?

Keep digging Jock.

27. phoodoo: Its the same idiocy that gives you ideas like some infinites are bigger than others, and Hilberts hotel can always find more space by having guests move over one room.

Have you published on these topics or is your contribution merely to label things as idiocy and leave your reputation to do the rest?

If the latter, well, you are fucked.

28. Err, can we agree on this, phoodoo:

If Billy is 25 meters along a 100m rope, and the rope is stretched one meter, then Billy would now be 25.25 meters from one end, and 75.75 meters from the other end.
If he moved 10cm during this stretch, he is either 25.15 / 75.85 from the ends, or 25.35 / 75.65, depending on which way he went.

?
If it makes you happier to include a 0.01cm stretch to Billy’s body, be my guest. Does not affect the big picture.

29. This is utterly amazing. You watched the video right? His body is not being stretched. He is being “carried along” in the direction of the stretch. The precise thing that you claim does not need to happen. It appears you don’t understand anything that is being discussed. Whatsoever.

Where is Graham2 now? He comes by with his little drive by thinking oh he is it figured out, and you have it figured out, and oh how dumb to think that it is the being carried along in one direction that is the key to the paradox, even though the narator himself says exactly that! I gave you the time stamp for crying out loud. Is he going to defend you now? Is anyone willing to defend you now, and say that you were right? I would love to see a poll, who is right, you or me?

I think you can get one consolation-Omagain will still believe you are right. Haha! Omagain! Sorry.

30. Oh, phoodoo, everyone is quite comfortable with the fact that you have been demonstrated to be hopelessly wrong.
There are all bored with your inability to understand anything.
I am only continuing as part of a psychological experiment, seeing how close I can lead you to water, without your ever drinking.
You were the one obsessing about his body getting stretched. Of course it’s immaterial.
Talking about Billy being “carried along” and “in the direction of the stretch” is dangerously vague, as Corneel explained to you.

Let’s apply some precision shall we?

Answer the “can we agree” questions — the most recent one (25meters out of 100) is the most basic one. If we cannot agree on that then there is no hope for you.
Then can I walk you through how and why you are wrong.

31. Let me just explain to you first, it actually gives me no great pleasure to tell you you are wrong-flat out wrong. I put this post here because I wanted to learn something. I was curious why no one else was seeming to understand the issue. So I wanted to hear what others thought. I was actually curious what your take was. But then I quickly realized you don’t have a clue. Add to that the typical arrogant manner in which you are so quick to think you are the only one who knows anything, and it makes it more palatable to tell you you are wrong. The paradox in this video only becomes a paradox becomes of the simple fact that “the ant is carried along” in the direction of the stretch. That is all it is. THAT’S IT. There is nothing else mysterious going on, no higher math needed, nothing. Its a trick. One that you are susceptible to.

Had you begun your response to me in a civil manner, I would have slowly, carefully explained to you what you aren’t getting. But no, you did what you always do. The whole trick to this video is that the ant is not only walking in the direction of the end at 5 cm/second, its is ALSO being carried in the direction of the end of the rope, which increases its distance. And this is why it can reach the end. If the rope was stretched from both ends, it would add 5cm to each end every second, and the ant would crawl five cm/s and it would never get anywhere. Its the carried along which gives him more progress than the 5 cm/s towards one end that it needs. Its as simple as that.

Its a lesson in not only paying attention, but also in realizing there are a bunch of idiots online. Don’t believe them. Yea, its me against ALL the other commenters on youtube who don’t get it. And you. And I am right. Thank you for making me realize there is nothing I am missing. If a bunch of people seem like they are acting like fools, it may well be that they actually are.

At some point a light is going to go on in your head, when you realize you are wrong. It probably already has, and that is why you are scrambling. You won’t admit it, because you have some ego issue. But learn from this Jock. I won’t expect you to admit it, but try to learn.

32. phoodoo: If the rope was stretched from both ends, it would add 5cm to each end every second, and the ant would crawl five cm/s and it would never get anywhere.

The joy, the pure unbridled joy!
So, to be clear, in phoodoo world:
Billy is 25 meters from the West end of a 100 meter rope, and the rope is stretched by 1 meter.
If the rope is anchored at the West end, and stretched at the East end, Billy is now 25 meters from the West end of a 101 meter rope.
If, on the other hand, the rope is anchored at the East end, and stretched at the West end, then Billy is now 26 meters from the West end of a 101 meter rope. (or perhaps the other way around, phoodoo has been unclear).
In either case, it is epic.
I encourage phoodoo to test his theory with a dot on a rubber band.

33. I have tried to make this easy for you, but that is a challenge.

Do the same mental experiment, but this time instead of have the ant attached to the band, have him hover one inch above it. Now each time the rope is stretched underneath him he does not get the benefits of progressing by the stretch. The only distance he gains is through walking. Now does he ever reach the end?

Sorry no he doesn’t. As long as the end he is trying to reach is traveling faster than him, he doesn’t get there. Please try to think a little longer OK?

Now you can pull both ends of the rope instead of one end, becausein that case the speed of each end is half of what it would be if you only pulled one end. So as long as one half of that speed is still as fast or faster than the ant, he doesn’t get there.

It’s amazing you don’t even get your own analogy. If you put a dot on a rubber band and stretch it, one half of the dot travels west and other travels east. It’s not making progress towards either end.

I know this hurts you because you are so proud of your phd and all. I am sorry a dumb rice farmer is making you look bad.

34. I have tried to make this easy for you,but that is a challenge.

Do the same mental experiment,but this time instead of have the ant attached to the band,have him hover one inch above it.Now each time the rope is stretched underneath him he does not get the benefits of progressing by the stretch.The only distance he gains is through walking.Now does he ever reach the end?

Sorry no he doesn’t.As long as the end he is trying to reach is traveling faster than him,he doesn’t get there.Please try to think a little longer OK?

Now you can pull both ends of the rope instead of one end, because in that case the speed of each end is half of what it would be if you only pulled one end. So as long as one half of that speed is still as fast or faster than the ant,he doesn’t get there.

It’s amazing you don’t even get your own analogy.If you put a dot on a rubber band and stretch it,one half of the dot travels west and other travels east.It’s not making progress towards either end.

I know this hurts you because you are so proud of your phd and all.I am sorry a dumb rice farmer is making you look bad.

35. Sorry, late to the discussion.

Phoodoo in the OP.

This is only true if the string is being stretched in only one direction, that is, the direction that it is approaching.

I’m not sure this is correct. All that matters is the distance between the two ends of the elastic and that the elastic is “ideal”. Imagining either end as a fixed point is simply changing the frame of reference.

Phoodoo, later:

phoodoo: Do the same mental experiment,but this time instead of have the ant attached to the band,have him hover one inch above it.

So, in what reference frame is the ant, now? Fixed, in relation to one or other end of the elastic? When it moves (assuming it can walk when hovering) its motion will be relative to what?

36. phoodoo: .If you put a dot on a rubber band and stretch it,one half of the dot travels west and other travels east.

It happens in the video, yes. But imagine the dot is small, the size of an atom. Atoms don’t stretch in the elastic (though maybe molecular kinks do). I’m confused as to whether we are supposed to be considering a mathematical paradox or reality.

37. Allan Miller: But what happens if you accelerate the stretching? In the example above, if you stretch at 1m/second/second, the marks outpace the ant after a couple of seconds. Is there an arbitrarily small subdivision in which the above reasoning holds? I’m not sure. It certainly seems intuitive that the acceleration will always outpace the ant eventually, however arbitrarily small you make the marks.

Right. That was going to be my next question. Moral: read earlier comments before commenting.

38. Hmmm. Suppose that the ant has on its back a massive sack of rice. It travels at 1cm/second. Each time it is 5mm from the previous grain of rice (measured using a handy laser sight), it dumps one on the rope. If the rope started out 1m long and wasn’t stretched, and the ant started off in the middle, it would clearly dump 100 grains before reaching the end.

Now we try stretching the rope, but reeeeeally slowly. If we stretch from both ends, with the ant in the middle, it is initially stationary in the reference frame that sees the ends moving apart at equal speeds, but as it moves away from the middle, it gains a component of motion from the stretch (in that reference frame). We’re stretching reeeeeally slowly here, so the ant can still readily get to the end. The distance between rice grains already dropped is expanding, due to uniform stretch. The longer ago a rice grain was dropped, the further from its neighbours it will become, even though they all started at 5mm. We’ll end up with more rice grains this way. The ant’s progress will be slower, for the same walking speed. But what won’t happen, from the ant’s pov, is a stationary pileup of grains. It can always get 5mm from the last grain, and so must be counting off subdivisions of the finite rope. ie, it must be getting somewhere.

Of course I’ve set the conditions such that the ant’s motion is much greater than the rate of stretch, making the stretch scenario little different from a rigid one. So the question is, as we move the ant back to the start and try stretching more quickly, is there a rate of stretch that sees the ant unable to get 5mm from its last grain?

On the relevance of which end(s) you stretch: Suppose, instead of stretching both ends equally, we stretch from one end. We have declared the other end as ‘fixed’, so that becomes the stationary point in a new reference frame. Now, with the ant in the middle, it moves relative to that fixed point from the outset, due to stretch alone. If it starts walking, its motion includes a component due to that stretch. There seems to be no difference between the two scenarios – tether one end or stretch both, provided the net amount of stretch is the same. Alternatively if you choose the ant’s pov, both ends are receding, however an outside observer might view the situation. As it moves from the middle, the end it is heading for is receding more slowly.

39. DNA_Jock … Your expression showed that the proportional distance must always increase, but does this mean the actual (remaining) distance must decrease ?

Later I may look at that, but I suspect that as the rope expands, it is possible for both to increase, ie: the ant advances proportionally, but the remaining distance also increases, ie: the ant retreats.

Maybe phoodoo is right.

40. I think the remaining distance the ant must travel after time t is:

x = (L + St) * (1 – (A/S) * ln (1 + St/L))

And as t increases it must eventually hit zero.
So phoodoo is wrong.

41. Graham,
Absolutely! The distance-to-go will always increase initially, in all scenarios where the result is paradoxical, i.e. where the ‘speed of stretch’ is greater than the speed of the ant. But, for all possible scenarios, the proportional distance is always increasing, and the ant will always reach the far end. It may take a very, very long time to get there.

For phoodoo’s benefit, here’s a discrete numerical treatment using the parameters in the video :

I am going to be deliberately conservative about the ‘help’ the ant gets.
L = 20 cm
S = 10 cm/s
A = 5 cm/s
After one second, the ant has moved 5cm down the rope, but the rope has stretched to 30 cm. Distance to Go (D2G) is 25 cm.
After the next second, the ant has covered 10 cm, and the rope is now 40 cm long…
BUT (whichever way one wants to measure it) the ant is an extra 10 x 5 / 30 cm (= 1.67 cm) from the west end due to the stretching, so he has in fact covered 11.67 cm, and D2G has increased to 28.33.
You might think it’s not looking good for the sad sack, but the thing is, he’s 11.67 / 40 of the way down the rope…
During the third second, then, he benefits from a further 2.92 cm of stretch, then 3.92, 4.75…
Six seconds in, he’s passed halfway — he still has over 31 cm to go, but he is benefiting from over 5 cm/s of stretch and (for these parameters) he’s closing inexorably on the east end. [His journey time will be less than 17 seconds under this approximation.]

Now, if you shorten the time periods to calculate what happens every tenth of a second, Billy will reach his goal sooner because I was deliberately conservative about the help, only applying the stretching benefit in the following time period…. (Journey time now 13.1 seconds).

Of course, as Graham notes, calculus yields the precise solution : t = L/S * (exp(S/A)-1) = 12.78 secs.
If Billy is slow, say 1cm/s, it will take him 12.24 hours.
If Billy also waits for ten minutes before he sets out, it will take him nearly six months.
But he always gets there eventually. This is highly counter-intuitive.

42. DNA_Jock: BUT (whichever way one wants to measure it) the ant is an extra 10 x 5 / 30 cm (= 1.67 cm) from the west end due to the stretching

Still don’t see the problem?

43. graham2: Later I may look at that, but I suspect that as the rope expands, it is possible for both to increase, ie: the ant advances proportionally, but the remaining distance also increases, ie: the ant retreats.

This is the case for a convergent series. That is, the ant would be asymptotically approaching some limit of proportional distance below 1. In the movie that phoodoo has linked to it is explained that for a harmonic series, which we happen to be dealing with here, this is not the case.

ETA: for extra irony points; the divergence of the harmonic series was already proven in the 14th century by Nicole Oresme, who besides math also studied economy, physics, astronomy, philosophy and theology. It’s people like him whom we have to thank for idiocy and dumb things like this.

44. The problem as stated above requires some assumptions to be made. The following fuller statement of the problem attempts to make most of those assumptions explicit

At time t=0 the rope starts to stretch uniformly and smoothly in such a way that the starting point remains stationary at x=0 while the target point moves away from the starting point with constant speed {\displaystyle v>0}v>0.

Duh!

Apologies unneccesary.

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