= Surface-fed growth of an accreting sphere
{title2=$r(t)=r_0+(\beta/\rho)t$}
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 $v=g[r-r_0^4/r^3]/(4\dot r)$.
Back to article page
