Ant On A Rubber Band

An ant is initially standing on the beginning of a kilometer long rubber band with that end fixed. at t = 0, the band begins to change its length by 1 km /s ( the band is a purely mathematical object - infinitely extensible ). At the same time, the ant begins crawling at 1 cm/s along the length of the rubber band ( on top of it ). How far has the ant traveled when he reaches the end?

@joe-shmo

Hi Joe

Unless I am being very dim, the end of this rubber band seems to be moving further away from the poor old ant as the seconds tick by??

Is it a trick question- the band goes around the earth and catches the ant up?

@blood-on-the-tracks said
@joe-shmo

Hi Joe

Unless I am being very dim, the end of this rubber band seems to be moving further away from the poor old ant as the seconds tick by??

Is it a trick question- the band goes around the earth and catches the ant up?

Your interpretation is correct in the sense that the end of the rubber band seems to be moving away from him at almost a kilometer a second. No trick question like curved geometry which it races along and catches up to the ant. The end of the band just keeps heading out in straight lines away from the ant.

Its not that you are being dim, its just that you need to right perspective to see what is happening.

Also: you need to consider the rubber band as having some "semi-real" properties of stretching ropes such that the volume of the rope is conserved ( i.e. matter isn't being created as its stretched, the rope is just thinning in lateral dimensions ). So perhaps i should have stated it is an infinitely voluminous rope to have this property of infinite extensibility at constant density.

Perhaps the first question is does the ant ever reach the end of the rubber band? Id recommend experimenting with dots on rubber bands to get a physical intuition of what might be the case.

@joe-shmo said
An ant is initially standing on the beginning of a kilometer long rubber band with that end fixed. at t = 0, the band begins to change its length by 1 km /s ( the band is a purely mathematical object - infinitely extensible ). At the same time, the ant begins crawling at 1 cm/s along the length of the rubber band ( on top of it ). How far has the ant traveled when he reaches the end?

So for every 100,000 centimetres the rubber band is extended by the ant gains 1 centimetre.
These numbers are on the large side so if we reduce them a bit:-
Assume the band is 10 cms long and extends at 10cms per sec.
The ant travelling at 1 cm per second gets to the end after 10 seconds.
Therfore this must also be true for 100,000 centimetres(or 1k) and 100,000 centimetres per second.
The distance travelled will be 1(velocity)=x/1000,000 = 100,000

@venda said
So for every 100,000 centimetres the rubber band is extended by the ant gains 1 centimetre.
These numbers are on the large side so if we reduce them a bit:-
Assume the band is 10 cms long and extends at 10cms per sec.
The ant travelling at 1 cm per second gets to the end after 10 seconds.
Therfore this must also be true for 100,000 centimetres(or 1k) and 100,000 centimetres per second.
The distance travelled will be 1(velocity)=x/1000,000 = 100,000

"Assume the band is 10 cms long and extends at 10cms per sec.
The ant travelling at 1 cm per second gets to the end after 10 seconds."

Examine this more carefully.

@joe-shmo said
"Assume the band is 10 cms long and extends at 10cms per sec.
The ant travelling at 1 cm per second gets to the end after 10 seconds."

Examine this more carefully.

After 1 second the band is 20cms long and the ant has travelled 11 cms
After 2 seconds the band is 30cms long and the ant has travelled 22 cms and so on, the ant gaining 1 cm per second.
Is this logic flawed?

@venda said
After 1 second the band is 20cms long and the ant has travelled 11 cms
After 2 seconds the band is 30cms long and the ant has travelled 22 cms and so on, the ant gaining 1 cm per second.
Is this logic flawed?

Yeah, im afraid so.
Try cutting a rubber band, place it next to a ruler measuring the initial length. Place an ink dot close to the fixed end ( noting the distance - and that the dot has no relative velocity with respect to the band ). Then double the length of the rubber band. Does the distance of the dot now increase to half the total length of the rubber band? Try moving the dot to different starting locations. closer to the end. What is happening?

@blood-on-the-tracks said
Unless I am being very dim, the end of this rubber band seems to be moving further away from the poor old ant as the seconds tick by??

Your're not very dim and everyone thinks that on seeing the puzzle for the first time!
It is a very old Maths "paradox" that needs some calculus to solve.

I think for this forum a fairer question is to prove he does reach the end.
(No calculus required.)

@joe-shmo said
An ant is

Very cruel Joe!
The question is normally how long will it take and I remember the answer being
the age of the Universe times 10000000000000000000000000000000000000000
😉

@wolfgang59 said
Your're not very dim and everyone thinks that on seeing the puzzle for the first time!
It is a very old Maths "paradox" that needs some calculus to solve.

I think for this forum a fairer question is to prove he does reach the end.
(No calculus required.)

To be fair, I did suggest ( a little bit ago ) that the initial stake in the ground should be convincing yourself the ant reaches the end...eventually!

🤫

@joe-shmo said
Yeah, im afraid so.
Try cutting a rubber band, place it next to a ruler measuring the initial length. Place an ink dot close to the fixed end ( noting the distance - and that the dot has no relative velocity with respect to the band ). Then double the length of the rubber band. Does the distance of the dot now increase to half the total length of the rubber band? Try moving the dot to different starting locations. closer to the end. What is happening?

Ok.Not much time at the moment but interesting

@blood-on-the-tracks said
@joe-shmo

Hi Joe

Unless I am being very dim, the end of this rubber band seems to be moving further away from the poor old ant as the seconds tick by??

Is it a trick question- the band goes around the earth and catches the ant up?

The distance the ant has travelled also increases as time goes by. The stretching does not happen all ahead of the ant. The area the ant already walked on stretches too so it looks like the ant is moving faster than 1cm/s from an observer stationary with respect to the start point.

@athousandyoung said
The distance the ant has travelled also increases as time goes by. The stretching does not happen all ahead of the ant. The area the ant already walked on stretches too so it looks like the ant is moving faster than 1cm/s from an observer stationary with respect to the start point.

Yes. Just as he reaches his destination he'll be at maximum velocity.
1.00001 km/s

3,600.036 kph!

Ok, so I think people are on board with the ant eventually reaching the end. We haven't shown an explicit proof, but if anyone wishes to answer the original question ( using Calculus as Wolfgang has pointed out ) have at it!