[Math] An Ant is Walking on a Rope...

[GodGenesis]

From where the stretch occurs does make alot of difference in this question also.

 

[Jei jei KK]

If each part stretches evenly then the ant will never reach the end.

How you're explaining this makes no sense though.

Think of it this way:

Let's assume the ant begins its journey 1/10th of the way in. Once the rope stretches, 900 meters is added to the other end and 100 to the other. There's now 200 meters behind the ant and 1800 in front of him. However, the relation of the ant's location with the rope stays the same; it is still 1/10th of the way in. No matter how small steps it takes, eventually it is going to end up halfway there, and eventually to the end.

Here's another example. Let's now change the ant's speed to a whopping two hundred meters per second:

0 seconds: Rope 1 kilometer, ant begins its journey.
1 seconds: The ant has traveled 200 meters. There is now 2 kilometers of rope, 200 behind the ant and 1800 in front of it. The ant has traveled 10% of the way.
2 seconds: The ant has traveled 400 meters. There is now 3 kilometers of rope, 400 behind the ant and 2600 in front of it.
3 seconds: 4 kilometers of rope, 600 behind the ant and 3400 ahead of it. The ant has traveled 15% of the way
4 seconds: 5 kilometers of rope, 800 behind the ant and 4200 ahead of it.
5 seconds: 6 kilometers of rope, 1000 behind the ant and 5000 ahead of it. 16.666...%
And so on. Relative to the rope, the speed of his progress gets slower and slower, but the ant will, given enough time (1.7 x 10^434 seconds if Imagine is right - I haven't bothered checking) reach the end of the rope.

 

[hugo cabret]

yes it will reach the end because the ant is moving at a rate of 1km and 1cm/ hour

 

[wackywamba]

It's natural to assume that the rope expands at 500 m/second at each end, so there's no way the ant can reach the end

Yes, 500m/s from the center point, but the ant isn't at the center point - it is continuously moving towards the end point.

That would mean it has a limit where it just can't stretch any more.

This has nothing to do with the solution.

 

[Noam]

Wait. In what direction is the rope stretched?

A, B, or C?

 

[Imagine]

Neither.

 

[Imagine]

Oh I get it, you don't add to it, you stretch it. That would mean it has a limit where it just can't stretch any more. I understand it but can't explain it

If you assume you are stretching it then surely it will break/not be able to stretch any further at a particular point, therefore the ant will reach the end.

We're assuming the rope will not break. To be honest, if we assumed the rope would break, then we'd also have to take into account that there would not be enough space (in the universe I believe) to let the rope expand to it's fullest (when the ant finally reaches the end).

If each part stretches evenly then the ant will never reach the end.

How you're explaining this makes no sense though.

No, if it stretches evenly, the ant will reach the end. You're just failing to understand what I'm saying. If it's 1/4 of the way done before the stretch, it'll be 1/4 of the way done after the stretch.

yes it will reach the end because the ant is moving at a rate of 1km and 1cm/ hour

I'm sorry, but how would the ant be moving at a rate of 1 km and 1 cm/h? That will never happen, even when the ant is at the very end...

Think of it this way:

Let's assume the ant begins its journey 1/10th of the way in. Once the rope stretches, 900 meters is added to the other end and 100 to the other. There's now 200 meters behind the ant and 1800 in front of him. However, the relation of the ant's location with the rope stays the same; it is still 1/10th of the way in. No matter how small steps it takes, eventually it is going to end up halfway there, and eventually to the end.

Here's another example. Let's now change the ant's speed to a whopping two hundred meters per second:

0 seconds: Rope 1 kilometer, ant begins its journey.
1 seconds: The ant has traveled 200 meters. There is now 2 kilometers of rope, 200 behind the ant and 1800 in front of it. The ant has traveled 10% of the way.
2 seconds: The ant has traveled 400 meters. There is now 3 kilometers of rope, 400 behind the ant and 2600 in front of it.
3 seconds: 4 kilometers of rope, 600 behind the ant and 3400 ahead of it. The ant has traveled 15% of the way
4 seconds: 5 kilometers of rope, 800 behind the ant and 4200 ahead of it.
5 seconds: 6 kilometers of rope, 1000 behind the ant and 5000 ahead of it. 16.666...%
And so on. Relative to the rope, the speed of his progress gets slower and slower, but the ant will, given enough time (1.7 x 10^434 seconds if Imagine is right - I haven't bothered checking) reach the end of the rope.

Exactly The series which you will be adding diverges to infinity, so given enough time, the ant will reach the end. And I may be wrong with the calculations, I tend to make stupid mistakes

 

[kill dank]

Think of it this way:

Let's assume the ant begins its journey 1/10th of the way in. Once the rope stretches, 900 meters is added to the other end and 100 to the other. There's now 200 meters behind the ant and 1800 in front of him. However, the relation of the ant's location with the rope stays the same; it is still 1/10th of the way in. No matter how small steps it takes, eventually it is going to end up halfway there, and eventually to the end.

Here's another example. Let's now change the ant's speed to a whopping two hundred meters per second:

0 seconds: Rope 1 kilometer, ant begins its journey.
1 seconds: The ant has traveled 200 meters. There is now 2 kilometers of rope, 200 behind the ant and 1800 in front of it. The ant has traveled 10% of the way.
2 seconds: The ant has traveled 400 meters. There is now 3 kilometers of rope, 400 behind the ant and 2600 in front of it.
3 seconds: 4 kilometers of rope, 600 behind the ant and 3400 ahead of it. The ant has traveled 15% of the way
4 seconds: 5 kilometers of rope, 800 behind the ant and 4200 ahead of it.
5 seconds: 6 kilometers of rope, 1000 behind the ant and 5000 ahead of it. 16.666...%
And so on. Relative to the rope, the speed of his progress gets slower and slower, but the ant will, given enough time (1.7 x 10^434 seconds if Imagine is right - I haven't bothered checking) reach the end of the rope.

explaining it like this, the ant would be the point at which the rope stretches. Behind the ant, the rope stretches backwords, in front of the ant the rope stretches forward. Yes he will keep progressing, but he will never get to the end because the rope is stretching at 1000 times the speed he's walking and always will be. Just because the rope behind the and stretches too, doesn't meanhe's traveled any further.


That's what makes logical sense. They way you're explaining it and looking at it doesn't.

 

[meerkatlady]

Thats mind boggling, maybe we should get an ant and prove it?? : )

 

[Imagine]

yes it will reach the end because the ant is moving at a rate of 1km and 1cm/ hour

explaining it like this, the ant would be the point at which the rope stretches. Behind the ant, the rope stretches backwords, in front of the ant the rope stretches forward. Yes he will keep progressing, but he will never get to the end because the rope is stretching at 1000 times the speed he's walking and always will be. Just because the rope behind the and stretches too, doesn't meanhe's traveled any further.


That's what makes logical sense. They way you're explaining it and looking at it doesn't.

He is travelling futher though... Another way to look at this is that the rope's expansion is linear while the ant's is exponential because the rope expanding pushes it forward more each time. (which means that the ant will catch up).

Thats mind boggling, maybe we should get an ant and prove it?? : )

As I said earlier, if we started at the time the universe started, the ant would not even be remotely close to reaching the end.

 

[Noam]

Neither.

I don't think you got my question.
Assume the ant is not moving at all.
If you stretch the rope like in fig. A, the and will receive a boost of 1km, because the entire stretching power occurred behind the ant.
In B, half the rope is stretched in front of the ant, and half behind it - effectively nulling any potential boost
In C, the entirety of the stretch occurs before the Ant, effectively puching it back 1km.

Assuming A is correct, the ant would reach the end as it advances 1.0001 km/s compared to the rope's effective -0.9999 km/s
Assuming B is correct, the ant would reach the end since it is advancing 1 cm/s as opposed to the rope's effective 0 cm/s
Assuming C is correct, the ant would never reach the end because it travels -0.9999 km/s, while the rope is expanding 1 km/s

 

[I'm Still Breathing]

I don't think you got my question.
Assume the ant is not moving at all.
If you stretch the rope like in fig. A, the and will receive a boost of 1km, because the entire stretching power occurred behind the ant.
In B, half the rope is stretched in front of the ant, and half behind it - effectively nulling any potential boost
In C, the entirety of the stretch occurs before the Ant, effectively puching it back 1km.

Assuming A is correct, the ant would reach the end as it advances 1.0001 km/s compared to the rope's effective -0.9999 km/s
Assuming B is correct, the ant would reach the end since it is advancing 1 cm/s as opposed to the rope's effective 0 cm/s
Assuming C is correct, the ant would never reach the end because it travels -0.9999 km/s, while the rope is expanding 1 km/s

I don't think you got his answer.

If the ant is 10% (1/10th) the way across the rope 10% of the stretch will happen behind the ant and the other 90% will happen in front of the ant, leaving the ant at 10% (1/10th) of the way across.

The ant will never lose distance percentage wise and as long as he keeps moving forward he will increase the percentage of the rope he has crossed, meaning eventually he will reach the end.

 

[Imagine]

I don't think you got my question.
Assume the ant is not moving at all.
If you stretch the rope like in fig. A, the and will receive a boost of 1km, because the entire stretching power occurred behind the ant.
In B, half the rope is stretched in front of the ant, and half behind it - effectively nulling any potential boost
In C, the entirety of the stretch occurs before the Ant, effectively puching it back 1km.

Assuming A is correct, the ant would reach the end as it advances 1.0001 km/s compared to the rope's effective -0.9999 km/s
Assuming B is correct, the ant would reach the end since it is advancing 1 cm/s as opposed to the rope's effective 0 cm/s
Assuming C is correct, the ant would never reach the end because it travels -0.9999 km/s, while the rope is expanding 1 km/s

Yeah, you didn't understand my answer. The picture I drew is a picture of one stretch, neglecting the distance the ant may have moved.

For example: Take a rubber band, cut it in half. (so it's just one long string). Mark a point on it. Now stretch the rubber band. Where is the point? It's the same situation with the ant.