Ant on a rubber rope Totally Explained

Everything about Ant On A Rubber Rope totally explained

Ant on a rubber rope is a mathematical puzzle with a solution that appears counter-intuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary,
but a typical form is as follows. ť An ant starts to crawl along a taut rubber rope 1km long at a speed of 1cm per second (relative to the rubber it's crawling on). At the same time, the rope starts to stretch by 1km per second (so that after 1 second it's 2km long, after 2 seconds it's 3km long, etc). Will the ant ever reach the end of the rope?

At first consideration it seems that the ant will never reach the end of the rope, but in fact it does (although in the form stated above the time taken is colossal). In fact, whatever the length of the rope and the relative speeds of the ant and the stretching, providing the ant's speed and the stretching remain steady the ant will always be able to reach the end given sufficient time.

A formal statement of the problem

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. ť Consider a thin and infinitely stretchable rubber rope held taut along an

x

-axis with a starting-point marked at

x=0

and a target-point marked at

x=c

c>0

.

ť 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

v>0

.

ť A small ant leaves the starting-point at time

t=0

and walks steadily and smoothly along the rope towards the target-point at a constant speed

alpha>0

relative to the point on the rope where the ant is at each moment.

ť Will the ant reach the target-point?

Solutions of the problem

An informal reasoned solution

If the speed at which the target-point is receding from the starting-point is less than the speed of the ant on the rope (for example, if

v), then it seems clear that the ant will reach the target-point (because it would eventually reach the target-point by walking along the axis, and walking along the rope can only carry it further forward).

Otherwise, we can still find a point on the rope that's receding at less than the speed of the ant on the rope by picking a suitable proportion of the distance from the starting-point to the target-point, for example

alpha/2v

of the way along (any amount less than

alpha/v

will work). Call this point

P_1

. It seems clear that the ant will reach

P_1

(because it would eventually reach

P_1

by walking along the axis, and walking along the rope can only carry it further forward). Now, the point on the rope at twice that proportion (call it

P_2

) is receding at exactly the same speed from

P_1

that

P_1

was receding from the starting-point (although it's by now rather further away). So the ant should be able to reach

P_2

. And now, the point on the rope at three times the proportion (call it

P_3

) is receding at exactly the same speed from

P_2

that

P_2

was receding from

P_1

(although it's much further away). So the ant should be able to reach

P_3

. This continues, and because the proportion of the way from the starting-point to the target-point at which each point,

P_1

P_2

P_3

, etc, is found is a fixed amount greater than the proportion for the previous point, the proportion will eventually reach and exceed 1, so the ant will eventually reach the target-point.
For the problem as originally stated, take

P_1

to be the point

1/200000th

of the way along the rope. This point is travelling away from the starting-point at half the walking-speed of the ant, so the ant has no trouble reaching it. Point

P_2

1/100000th

of the way along the rope and is travelling away from

P_1

at half the walking-speed of the ant,

P_3

3/200000ths

of the way along, etc, so after repeating the achievement 200,000 times the ant reaches the end of the rope. However, as the distance gets longer each time, so the time to complete each

1/200000th

of the way gets longer each time, it's clear that the time required for the ant to complete the journey will be very large. This solution doesn't provide any more precise indication of how long it'll take.

A discrete mathematics solution

Although solving the problem appears to require analytical techniques, it can actually be answered by a combinatorial argument by considering a variation in which the rope stretches suddenly and instantaneously each second rather than stretching continuously. Indeed, the problem is sometimes stated in these terms, and the following argument is a generalisation of one set out by Martin Gardner, originally in Scientific American and later reprinted. Consider a variation in which the rope stretches suddenly and instantaneously before each second, so that the target-point moves from

x=c,!

x=c+v,!

at time

t=0,!

, and from

x=c+v,!

x=c+2v,!

at time

t=1,!

, etc. Many versions of the problem have the rope stretch at the end of each second, but by having the rope stretch before each second we've disadvantaged the ant in its goal, so we can be sure that if the ant can reach the target-point in this variation then it certainly can in the original problem or indeed in variants where the rope stretches at the end of each second.
Let

heta(t),!

be the proportion of the distance from the starting-point to the target-point which the ant has covered at time t. So

heta(0)=0,!

. In the first second the ant travels distance

alpha,!

, which is

frac,!

. This is a truly vast timespan, vast even in comparison to the estimated age of the universe, and the length of the rope after such a time is similarly huge, so it's only in a mathematical sense that the ant can ever reach the end of this particular rope.

Applications of the problem

This problem has a bearing on the question of whether light from distant galaxies can ever reach us if the universe is expanding. 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 doesn't violate the relativistic constraint of not travelling faster than the speed of light, because the galaxy isn't "travelling" as such -- it's 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.

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