An ant starts at one end of a rubber band 1 km long and crawls toward the far end at 1 cm per second. The band is stretched uniformly and continuously at 1 km per second, carrying the ant with it. Does the ant ever reach the far end, and if so when?

An ant starts at one end of a rubber band 1 km long and crawls toward the far end at 1 cm per second. The band is stretched uniformly and continuously at 1 km per second, carrying the ant with it. Does the ant ever reach the far end, and if so when?

Approach: Track the fraction of the band behind the ant rather than its absolute position, since uniform stretching leaves that fraction unchanged.

e^{100000} - 1 seconds. Let f be the fraction of the band behind the ant. Uniform stretching carries the ant with the material and leaves f unchanged, so f grows only from crawling, giving the separable differential equation df/dt = v/L(t) with v = 1 cm/s and L(t) = 10^5 + 10^5 t cm. Integrating gives f(t) = (1/10^5) ln(1 + t), and f = 1 when ln(1 + t) = 10^5, so t = e^{100000} - 1 seconds. The ant does arrive, because the harmonic divergence of the integral of 1/(1 + t) means f is unbounded. The instinct that the far end runs away faster than the ant crawls fails because the stretch carries the ant forward as well, and only the remaining fraction matters.

Follow-up: If the band instead stretches by a fixed 1 km at the end of each second, does the arrival time change in order of magnitude?

Key concepts: fraction of the band, harmonic divergence, separable differential equation.