The Ant on a Rubber Rope Paradox

An ant is inching along a rope at one centimeter per second. The rope is made of rubber, and being stretched at a rate of kilometer per second. Will the ant ever reach the end? Here's a paradox for people working on long, arduous projects.

One of the more puzzling paradoxes out there is the ant on a rubber rope paradox. It goes by many names, sometimes being referred to as the inchworm on a rope paradox, but the exact circumstances don't matter. It always seems impossible.

An ant is placed on the very end of a one meter long rubber rope. At the far end of the rope is a car. The moment the ant starts moving, the car takes off. The car moves at one kilometer per second. The ant moves at one centimeter per second. Will the ant ever reach the end of the rope? It seems impossible. The rope is stretching at a faster rate than the ant is moving. How could it possibly catch up? A normal ant could not, it's true. We have to assume an immortal ant, an unending supply of fuel, an infinitely stretchable rope that stretches uniformly throughout its length, and, for that matter, an unending universe. If we gather all those elements together, the ant will, eventually, get to the end.

The problem seems impossible because we picture the ant and the rope moving independently. When you realize that the ant is on the rope, and the portion of the rope behind the ant stretches just as the portion in front of the ant does, things become a little more imaginable. The mathematics are complex, but think of the entire picture of the ant and the rope. At zero seconds, the ant is at the very end of the rope, and has 100 percent of the rope in front of it. At one second, it's true that the ant's task is considerably increased, but it doesn't have one hundred percent of the rope in front of it. And that tiny percentage of the rope that the ant has traversed will stretch in proportion, just like the rest of the rope. Instead of thinking of the ant as falling farther and farther behind the car, think of it as gaining slowly increasing percentages of the rope. Eventually, the percentage will shrink to nothing.

In this case, it seems it will shrink to nothing after 2.8 x 10^43,429 seconds, but hey, keep walking little ant!

[Via Ant on a Rope.]