Chapter 8, Problem 8.103CP

CALC A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is

$F_{\text{ext}}=\frac{dp}{dt}=m\frac{d\upsilon }{dt}+\upsilon \frac{dm}{dt}$

Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m = kx, where k is a constant, and dm/dt = kυ. This gives, since F_ext = mg,

$mg=m\frac{d\upsilon }{dt}+\upsilon (k\upsilon )$

Or, dividing by k,

$xg=x\frac{d\upsilon }{dt}+{\upsilon }^2$

This is a differential equation that has a solution of the form υ = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero, (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the rain-drop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113–117.)