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 Michael A. Gottlieb
Assume that the bug's distance from the wall when the band is not stretched is f(t) and the length of the rubber band (when it is stretched) is r(t). Let the distance from the wall to the bug (crawling along the band as it is stretched) be b(t). Then for an infinitesimal period dt, we have
b(t+dt) = b(t)r(t+dt)/r(t) + df(t).
Setting r(t+dt) = r(t) + r'(t)dt and df(t) = f'(t)dt, rearranging,
(b(t+dt)-b(t))/dt = b(t)r'(t)/r(t) + f'(t).
Taking the limit as dt approaches 0,
b'(t) = b(t)r'(t)/r(t) + f'(t).
The solution to this differential equation with boundary condition b(0)=0, is
b(T) = r(T)(Integral t=0->t=T f'(t)/r(t) dt),
If the bug reaches the end of the band at time T, so that b(T)=r(T), then we have
Integral t=0->t=T f'(t)/r(t) dt = 1.
The solution in T of this equation (when one exists) gives the answer for the general case.
Let f(t) = 0.00001t, and r(t) = t+1. Then we have
Integral t=0->t=T 0.00001/(t+1) dt = 1,
or, carrying out the integral,
0.00001 ln(T+1) = 1.
So, T = e^100000 - 1 seconds.