A spherical raindrop of radius $a(t)>0$ and density $\rho$ falls down at a velocity $v(t)>0$ through a fine stationary mist. As the raindrop falls its volume grows at the rate $c \pi a^{2} v$ with constant $c$. The raindrop is subject to the gravitational force and a resistive force $-k \rho \pi a^{2} v^{2}$ with $k$ a positive constant. Show $a$ and $v$ satisfy

$$\begin{aligned} &\dot{a}=\frac{1}{4} c v, \\ &\dot{v}=g-\frac{3}{4}(c+k) \frac{v^{2}}{a} . \end{aligned}$$

Find an expression for $\frac{d}{d t}\left(v^{2} / a\right)$, and deduce that as time increases $v^{2} / a$ tends to the constant value $g /\left(\frac{7}{8} c+\frac{3}{4} k\right)$, and thence the raindrop tends to a constant acceleration which is less than $\frac{1}{7} g$.