A bug walks down a rubber band which is attached to a wall at one end and a car moving away from the wall at the other end.

analysis/rubberband Added material

Problem

A bug walks down a rubber band which is attached to a wall at one end
and a car moving away from the wall at the other end. The car is moving
at 1 m/sec while the bug is only moving at 1 cm/sec. Assuming the
rubber band is uniformly and infinitely elastic, will the bug ever
reach the car?

Show solution

Let w = speed of bug and N = ratio of car speed/bug speed = 100.  Paint
N+1 equally spaced stripes on the rubberband.  When the bug is standing
on one stripe, the next stripe is moving away from him at a speed
slightly < w (relative to him).  Since he is walking at w, clearly the
bug can reach the next stripe.  But once he reaches that stripe, the
next one is only receeding at < w.  So he walks on down to the car, one
stripe at a time.

The bug starts gaining on the car when he is at the next to last
stripe.

Additional material

[ Editor: a January 1997 posting on a version with a truck at 100 mph,
an insect walking at 1 mph, and a band initially 10 inches long. ]

Let L be the initial length of the rubber band, v the insect's walking
speed, and w the truck's speed.  If the rubber band stretches
uniformly, its starting point is at x = 0, and it starts to stretch at
time t = 0, then the velocity of the point x on it at time t is
x*w/(L+w*t).  At any given time t the insect's velocity is thus
v+x(t)*w/(L+w*t), where x(t) is the insect's position at this time.
The insect's velocity is the time derivative of its position, and thus

x'(t) = v + x(t) * w / (L+w*t)

with the boundary condition

x(0) = 0.

The solution of this differential equation is given by

x(t) = (v/w) * log(1+w*t/L) * (L+w*t).

The truck's position at time t is L+w*t, and thus the insect reaches
the truck when (v/w)*log(1+w*t/L) = 1, i.e. when

t = (L/w) * (exp(w/v)-1)

if L = 10", v = 1 mph and w = 100 mph, using the above formula yields
t = 1.5e+41 seconds = 4.8e+33 years.

-- Ilan Mayer <ilan@isgtec.com>