Ant on a rubber rope

  • Thread starter AlexGTV
  • 58 comments
  • 5,363 views

Will the ant ever reach the ending point of the rope

  • Yes

    Votes: 24 60.0%

  • No

    Votes: 16 40.0%
  • Total voters
    40
I think the discrete math solution given in the wiki article is the best way to understand this problem conceptually (even if you haven't studied discrete). This type of problem is pretty common in discrete math courses. It's not really necessary to bother with all the calculus and logarithms involved in calculating an actual answer, especially because the amount of time needed by the ant is incredible anyway.
 
Bram Turismo
This depends on where you place the stretching point on the rope. The ant is moving with the rope if the rope is being held at its starting point. But if one assumes the midpoint of the rope along its length is the starting point, then the ant will actually be removed further from the endpoint of the rope in comparison to when the rope is not stretched.

If the rope is held in its middle, I think it's all a question of how long the rope can be extended until it snaps. Also don't forget ants have a certain lifetime. If one assumes the rope can be stretched indefinitely or not, one also has to assume how long an ant lives.
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I can see where you are getting confused, I struggled to get my head around it for quite a while. The answer is, it doesn't matter where it is being stretched from (whether it be from one end or both ends), if the rope is stretched, then the ant will move along the rope with it.

Let's say for instance, that we put a mark with a pen on the rope 1m from the left hand end of the rope. If we then stretch the rope 1km, so that it doubles in length, does that mean that the mark is still 1m from the end, or now 2m from the end? If you suggest that it is still 1m from the end of the rope, then you have not stretched the rope, instead you have extended the length of the rope. Note, this is not the same as stretching it. If we then stretch the rope another 1km, not extend, then the mark will now be 3m from the left hand end.

If you don't believe me, try this with an elastic band and two screws. Take a 5cm diameter elastic band and mark it three times, two directly opposite each other, and another mark 1cm anti-clockwise from one of the marks (preferably facing towards you). Screw the two screws into a piece of wood 5cm apart, lining up the marks so they are opposite each other on the screws, and measure the mark on the elastic band. You should find that it will be 1cm from the left hand screw. Now take the right hand screw out and move it another 5cm to the right (so now it is 10cm from the one on the left). Align the marks, and you will find the third mark is now 2cm from the one on the left. Note, you haven't extended the original length of the elastic band, merely increased the distance covered by the material. As you increase the material, so you increase the surface, and it's this that makes the ant move along the rope faster than the 1cm/s it can travel under it's own power.

So, how does this answer your points? While it is true that the endpoint is further away from the ant with each passing second in terms of distance, the percentage of the total distance actually decreases as it moving along the rope at 1cm/s plus a fraction of the total movement of the rope. When the ant reaches the relative midway point of the rope, it will actually start to get the end of the rope faster and faster (relatively speaking) than it was before it reached the midway point.

As for your second point, this experiment is a hypothetical one, requiring an elastic rope that will never break, and an ant that will last eternity. As for how long it would take, I would take a guess at around a couple of hundred million years.
 
After a while the ant will reach the original 1km mark on the rope due to it being stretched even if the ant didn't cover that much distance.
 
:drool:.........................
i-have-no-idea-what-im-doing.jpeg
 
kennylmao
Great question that is, and has got me thinking.

If you minus 10% from any number and continue to do the same(to minus 10%) for the result of the initial action, would the result eventually be 0(zero)? :dopey:

What do you think? DISCUSS! :)
Click to expand...

No, you end up with an infinite number of decimal places, but you would still end up with a number greater than 0.
 
DG_Silva
If we then stretch the rope 1km, so that it doubles in length, does that mean that the mark is still 1m from the end, or now 2m from the end? If you suggest that it is still 1m from the end of the rope, then you have not stretched the rope, instead you have extended the length of the rope.
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I never suggested so.

So, how does this answer your points? While it is true that the endpoint is further away from the ant with each passing second in terms of distance, the percentage of the total distance actually decreases as it moving along the rope at 1cm/s plus a fraction of the total movement of the rope. When the ant reaches the relative midway point of the rope, it will actually start to get the end of the rope faster and faster (relatively speaking) than it was before it reached the midway point.
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And we disagree where, exactly?

Note by saying:

But if one assumes the midpoint of the rope along its length is the starting point, then the ant will actually be removed further from the endpoint of the rope in comparison to when the rope is not stretched.
Click to expand...

I think of the middle point at which it is stretched as a point being held against movement. So obviously either side of the point which is the stretch-center would stretch in opposite sides to one another. In this case I think of the ant walking on the left side of the stretch point, while its destination lies on the right side.
 
Bram Turismo
I think of the middle point at which it is stretched as a point being held against movement. So obviously either side of the point which is the stretch-center would stretch in opposite sides to one another. In this case I think of the ant walking on the left side of the stretch point, while its destination lies on the right side.
Click to expand...

Which is the same as 2x 500m ropes tied together. It wouldn't make a difference, it would get the the midpoint and then do the same on the second rope. It would take longer on the second rope (After the midpoint) as it will have already been stretched, but it would get to the end eventually.
 
I was doing some maths last night, and lost the plot. Typing in this: =(2.8*(10^20))/3155760 into a spreadsheet gives 88,726,645,879,281.1 years. Spreadsheet won't even entertain the idea of 10^43429.
 
DG_Silva
Hmmm, I read the article after I answered. I cannot understand how the ant can move along an ever expanding rope when the ending position is further away in a constant extension. Is there a simpler way of explaining it without resorting to vast amounts of algebra?
Click to expand...

That isn't algebra really. Well, it has some of it in there.
 
This all seems like an awful lot of effort to expend tracking bugs. :odd:

But really, I have no idea how one would go about scientifically solving the problem, save for the fact that in real terms, the time needed for the ant to get to the end would exceed the lifespan of the ant many, many times over. And more likely, any normal rope would break after a fraction of a second.

It's an interesting, if impracticable theory.
 
AlexGTV
From the wiki article of the same name:

"An ant starts to crawl along a taut rubber rope 1 km long at a speed of 1 cm per second (relative to the rubber it is crawling on). At the same time, the rope starts to stretch by 1 km per second (so that after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc). Will the ant ever reach the end of the rope?"

It is easy to picture, but can you give a definite answer?

Vote and then check below.

The answer in the link:
http://en.wikipedia.org/wiki/Ant_on_a_rubber_rope
Click to expand...

Animal cruelty




Reported.
:P
 
Moglet
Animal cruelty




Reported.
:P
Click to expand...

:)

Wow, I'm glad you never saw me and my friends dumping ants into a metal wagon to which we connected one side of a Model T spark coil. The other side was connected to a stout screwdriver with a thick plastic handle, allowing us to bring down the lightning onto our chosen victims. :crazy:
 
Well, we had some stuff like this in our math class. Instead, we just used cars as an example. It's been a few years, but I kinda remember how it goes. Car A is in the lead, but car B is driving faster. Simple math dictates that the distance between the two cars is constantly decreasing, but it will get infinitely small instead of becoming negative.

There is some sort of solution for this, but I completely forgot about it.

Anyways, the ant's going to die before it reaches the end of the rope, anyways. If there was a rope that was able to stretch like that. And there isn't.

What do we learn from that? Reality > math.
 
Yeah, so this has been bugging, trying to ignore the distractions of ant and rubber, and just take the problem for what it is, I still say no, it will not.. which is probably contradictory to what much more intelligent people than me say but.. here goes. (I've don'e most of it in metres)

At t=0 (before any time has elapsed), the ant is at d=0 (d being the distance from the start point), and e=1000m (1km... e=the end point) and L=1000m(length or rope).

At t=1 the Ant has travelled 0.01m (1 cm) under it's power, and since L has now doubled to 2000m the length of the 0.01m the ant managed on it's own has also doubled, meaning that the ant has actually travelled 0.02m from the start point, meaning that the end point is 1999.98m away.

At 2 seconds in (t=2), the ant has travelled another 0.01m on it's own, the value of that 0.01 has been stretched again, since L is now 3000m. The extra 1000m is split up between the existing 2000m, meaning this 0.01m is only worth half again, so 0.015 all up for the ant, taking d to a total of 0.035m (0.02m+0.015m), but since L is now 3000m, e now equals 2999.965m

Let's not forget that the any is still only travelling at 1cm per second relative to the exact point of rubber he happens to be above at that point.

So, at t=3... L=4000m, the ant's centimetre is now worth 0.0133333m, so adding that to it's d so far give us...0.048333 and e = 3999.951667




...



at t=22552 seconds, the extension of the rope is only adding microns on to the 1cm/s that the ant is travelling, but it's still adding 1000m to the distance it needs to travel...

errrm...


The fastest the ant will ever travel away from the start point is 0.02 metres per second, from then on it's speed only decreases, getting closer to 1cm/s in infinatley smaller steps... the end point of the rope is still travelling away at 100,000cm/s.. and not decreasing.


errm...

this assumes the extra distance is spread evenly across the entire length of the rope,..

hmm.. I'm confused now.
 
Luminis
Well, we had some stuff like this in our math class. Instead, we just used cars as an example. It's been a few years, but I kinda remember how it goes. Car A is in the lead, but car B is driving faster. Simple math dictates that the distance between the two cars is constantly decreasing, but it will get infinitely small instead of becoming negative.

There is some sort of solution for this, but I completely forgot about it.

Anyways, the ant's going to die before it reaches the end of the rope, anyways. If there was a rope that was able to stretch like that. And there isn't.

What do we learn from that? Reality > math.
Click to expand...

Then maybe the practical application at the end of the wiki article would interest you...

This problem has a bearing on the question of whether light from distant galaxies can ever reach us if the universe is expanding.[2] If the universe is expanding uniformly, this means that galaxies that are far enough away from us will have an apparent relative motion greater than the speed of light. This does not violate the relativistic constraint of not travelling faster than the speed of light, because the galaxy is not "travelling" as such—it is the space between us and the galaxy which is expanding and making new distance. The question is whether light leaving such a distant galaxy can ever reach us, given that the galaxy appears to be receding at a speed greater than the speed of light.

By thinking of light photons as ants crawling along the rubber rope of space between the galaxy and us, it can be seen that just as the ant will eventually reach the end of the rope, given sufficient time, so the light from the distant galaxy will eventually reach Earth, given sufficient time.
 
DG_Silva
No, you end up with an infinite number of decimal places, but you would still end up with a number greater than 0.
Click to expand...

If the number is 0.000... with an infinite number of decimal places, it is by definition equal to zero.

At first I thought this puzzle had to do with time tending towards infinity, travelling 1cm/s for infinity is equal to travelling 1km/s for infinity, but once I had read this thread, it does start to make sense.

I suppose it's one of those things that you either understand, or can't comprehend (no offense intended)
 
Luminis
Reality > math.
Click to expand...

Well, on the other hand based on quantum mechanics it is the other way around. Or we can say known reality < math. No one understand quantum mechanics very well, wave-particle duality e.t.c. but the math are the most successful of any physics theory ever!~
 
AlexGTV
Well, on the other hand based on quantum mechanics it is the other way around. Or we can say known reality < math. No one understand quantum mechanics very well, wave-particle duality e.t.c. but the math are the most successful of any physics theory ever!~
Click to expand...
Given that we are using math to explain what we are seeing in reality, I stand by my point. Math is used to depict and explain reality, not the other way around.
 

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