bug on band An infinitely stretchable rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s;  initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?

Solution by Rudolf Pfeiffer

Let the fixed end of the band be at the origin. The right end moves with velocity v0 such that the end is at position L0 + v0 * t.

The bug moves with velocity v relative to the band. However, for a still standing observer it has a higher velocity. This can most easily be seen by setting v = 0. Since the band stretches, even a bug standing on the band would move relative to a still standing observer and it would move the faster the further it is away from the fixed end of the band. Thus, the still standing observer gets the following differential equation for the position of the bug (' denotes the time derivative):

x' = x/(L0 + v0 * t) * v0 + v with x(0) = 0.

Obviously, as the bug moves to the right (relative to the still standing observer) its velocity increases and as soon as it is larger than v0, the distance of the bug to the right end begins to decrease.

The solution to the DE is

x(t) = v/v0 * (L0 + v0 * t) * ln (L0 + v0 * t)/L0

Setting this equal to L0 + v0 * t gives the time when the bug reaches the right end of the band:

t = L0/v0 * [exp(v0/v) - 1]

The bug reaches the velocity v0 after the time

t = L0/v0 * [exp(v0/v - 1) - 1]

So even if v is many magnitudes smaller than v0, the bug will reach the end of the band in a finite time. With L0=1 m, v0=1 m, and v = 0.00001 m/s, t = e^100000 - 1.