Solved: Raindrops Keep Falling In meteorology the term

Chapter 1, Problem 1.1.147

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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 evaporatesthat is, the rate at which it loses massis 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 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 0. Ignore air resistance. [Hint: Use the form of Newtons second law given in (17).]