Raindrops Keep Falling In meteorology the term virga refers to falling raindrops or...

Raindrops Keep Falling In meteorology the term virga refers to falling raindrops or ice particles that evaporate before they reach the ground. Assume that a typical raindrop is spherical. Starting at some time, which we can designate as t = 0, the raindrop of radius r₀ falls from rest from a cloud and begins to evaporate.

(a) If it is assumed that a raindrop evaporates in such a manner that its shape remains spherical, then it also makes sense to assume that the rate at which the raindrop evaporates—that is, the rate at which it loses mass—is proportional to its surface area. Show that this latter assumption implies that the rate at which the radius r of the raindrop decreases is a constant. Find r(t). [Hint: See Problem 51 in Exercises 1.1.]

(b) If the positive direction is downward, construct a mathematical model for the velocity v of the falling raindrop at time t. Ignore air resistance. [Hint: When the mass m of an object is changing with time, Newton’s second law becomes , where F is the net force acting on the body and mv is its momentum.]