Assignment:
Question 1. A raindrop is observed at time t=0 when it has mass m and downward velocity u. As it falls under gravity its mass increases by condensation at a constant rate and a resisting force acts on it, proportional to is speed and equal to v when the speed is v.
a) Show that d/dt((M^2)v) = (M^2)g where M = m+ t.
b) Show that the speed of the raindrop at time t is (g/3 )[m+ t-((m^3)/(m+ t)^2)] + ((m^2)u)/(m+ t)^2
Question 2. A particle falls from rest under gravity through a stationary cloud. The mass of the particle increases by accretion from the cloud at a rate which at any time is mkv, where m is the mass and v the speed of the particle, and k is a constant.
a) Show that after the particle has fallen a distance x, kv^2 = g(1-e^(-2kx)).
b) Show that the distance the particle has fallen after time t is (1/k)ln(cosh[t(g/k)^0.5)]