Chapter 3.1, Problem 41E

Evaporating Raindrop As a raindrop falls, it evaporates while retaining its spherical shape. If we make the further assumptions that the rate at which the raindrop evaporates is proportional to its surface area and that air resistance is negligible, then a model for the velocity v(t) of the raindrop is

$\frac{dv}{dt}+\frac{3(k\text{/}\rho )}{(k\text{/}\rho )t+r_0}v=g$.

Here ρ is the density of water, r₀ is the radius of the raindrop at t = 0, k < 0 is the constant of proportionality, and the downward direction is taken to be the positive direction.

1. (a) Solve for v(t) if the raindrop falls from rest.
2. (b) Reread Problem 36 of Exercises 1.3 and then show that the radius of the raindrop at time t is r(t) = (k/ρ)t + r₀.
3. (c) If r₀ = 0.01 ft and r = 0.007 ft 10 seconds after the raindrop falls from a cloud, determine the time at which the raindrop has evaporated completely.