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

An ant is walking on a rope 1 km long at a speed of 1 cm/second. The rope is also being stretched at a rate of 1 km/second. (The ant will move with the rope when it is stretched).

Will the ant ever reach the end?

If so, explain why or why not.


(Just for the record, I know the answer and I have a mathematical proof of yes/no. I just thought this problem was rather interesting).

 

My gut is telling me that it will reach the end... As the ant moves, some of the rope ends up behind it. That means some of the stretching will occur behind the ant making the ant "move" further towards the end. Or maybe it will approach a certain amount across but never reach it. I really have no idea.

 

Some of the rope ends up behind it, that's true. But does the whole 1 km end up behind it? (only when it is at the very end...)

EDIT: I may be directing or misdirecting you, I'm just going to attempt to explain why a certain solution wouldn't work... etc, if it makes sense.

 

I know the whole 1km does not end up behind it. Which is why I proposed the "it will approach a point but never reach it". I really dunno so idk.

 

Just by doing a quick read over the question, my guess is that it will never reach the end of the rope.

 

You guys can PM me if you want to know the answer. Otherwise, I'll just post it here later.

 

He will never reach the end because no rope has the tensile strength to be infinitely stretched, so therefore the rope will snap and the ant will fall to the ground.

 

Glancing over it and doing no math, the ant will reach the end. The only problem is when? It will take a very very very large amount of time because his distance traveled gets larger and larger..Pm me the answer I think I am right though.

Edit: This pissed me off and I did some math. He will reach the end but it is not EXACTLY like I explained earlier.

 

The growth of the length of the rope is linear. The speed that at which the ant transverses the rope is exponential. This is because although the rate of walking is linear, the amount of rope behind him continues to grow. The more rope behind him, the farther he goes each second. Since exponential things grow faster than linear things (if you extend to infinity) the ant most definitively reaches the end at some point.

 

Solving it now. It's a paradox, which I'll post the name of later. .
I couldn't find the solution at this moment. I'll try to do it later though. D:

 

Never mind. Math time later

 

it will reach the end.
pm me solution

 

Alright since I've received a couple of requests for the solution, I'll post it here.

The ant will reach the end.

Essentially, think of this as an infinite sum. You have the ant traveling 1/1000 of the distance the first second, + 1/2000 the second, + 1/3000, + 1/4000, etc. Essentially, you can factor out 1/1000, and you have the summation of (damn forums, no math equation tools) 1/2 + 1/3..., which diverges to infinity. So essentially, the ant is capable of traveling any amount of distance.

An extra thing to note though would be the amount of time it takes to travel. If you do some further calculations, you can figure out that the time it takes to reach the end is:

which is approximately

It is interesting to note that the universe has only been around for

, so try to grasp the huge amount of time it would take.

 

That's a pretty cool equation, sythe needs more of these maths q's.

 

It depends where the rope is stretched. If the end behind the ant stretches, it will always take the same time for the ant to get to the end. If it's in front of him, he will never reach the end because the rope will be getting 100,000cm longer every time the ant travels 1cm.

 

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

 

I included the solution. The ant will reach the end. When a rope stretches uniformly, every part of the rope stretches. I.e. if the rope is 1 km long, and you stretch it by 1 km, the part that was @ 1 cm before is not @ 2 cm.

^Same as Kill dank, you don't just add 500 m/s at each end, you stretch it by 1 km/s.

 

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

How you're explaining this makes no sense though.

 

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.