In a rocketpropulsion 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

$$F_{ext} = {dp\over dt} = m {dv\over dt} + v {dm\over dt}$$

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

$$mg = m {dv \over dt} + v(kv)$$ Or, dividing by $k$, $$xg = x {dv\over

dt} + v^2$$

This is a differential equation that has a solution of the form $v = at$, 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 raindrop 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.)