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 Twan Mennink
Introduce a coordinate system where zero is the beginning of the band and 1.0 is the total length of the band. As the band is expanding over time, the coordinate system is also expanding. The speed of the bug at time zero is 0.00001 km/s. The speed of the bug at time t expressed in this expanding coordinate system is 0.00001/(1 + t). We want to know if and when the bug reaches the end of the band, which is position 1.0 in the expanding coordinate system, so we have to integrate the speed from t=0 to t=T and solve the equation for T:
Integral from t=0 to T of (0.00001/(1 + t )) = 1.0,
Integral from t=0 to T of (1/(1+t)) = 100000,
which gives ln(1+t)=100000. So, t = exp(100000) -1.