Chapter 9.3, Problem 49E

When a raindrop falls, it increases in size and so its mass at time t is a function of t, namely, m(t). The rate of growth of the mass is km(t) for some positive constant k. When we apply Newton’s Law of Motion to the raindrop, we get (mv)′ = gm, where v is the velocity of the raindrop (directed downward) and g is the acceleration due to gravity. The terminal velocity of the raindrop is lim_t→∞v(t). Find an expression for the terminal velocity in terms of g and k.