The ant on a rope that keeps stretching

Question

An ant starts at one end of a rubber rope that is 1 km long. It crawls towards the other end at 1 cm per second. At the end of every second the rope is stretched uniformly so that it becomes 1 km longer: after one second it is 2 km long, after two seconds 3 km, and so on. The stretching carries the ant along with it, in proportion to where it is on the rope. The rope can stretch forever and the ant lives forever.

Does the ant ever reach the far end of the rope? If so, roughly how long does it take?

Accepted Answer

Yes, it reaches the end, after an astronomically long but finite time: roughly e to the power 100,000 seconds. Fractions, not distances. When the rope stretches uniformly, every point moves in proportion to its position, so the fraction of the rope behind the ant is unchanged by stretching. Only the ant's own crawling changes that fraction. What crawling achieves. During the first second the rope is 1 km = 100,000 cm long, so crawling 1 cm covers 1/100,000 of the rope. During the second second the rope is 200,000 cm long, so 1 cm covers 1/200,000. During the n-th second it covers 1/(100,000 n…

The ant on a rope that keeps stretching

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