 Surface-fed growth of an accreting sphere

For a sphere with constant mass density $\rho$ collecting mass at rate $\beta$ per unit surface area, $\dot{m}=4\pi r^2\beta$ and $m=4\pi \rho r^3/3$ give $\dot{r}=\beta /\rho$. If it falls through stationary material under constant gravitational acceleration $g$ without drag, its velocity satisfies $\dot{v}+3\dot{r}v/r=g$. With $v(0)=0$, an integrating factor $r^3$ gives $v=g[r-r_0^4/r^3]/(4\dot{r})$.