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The mass and surface area of the spherical droplet are $m=(4\pi /3)\rho r^3$ and $A=4\pi r^2$. Constant surface mass accretion rate $\beta$ meansTherefore $\dot{r}=\beta /\rho =\mu$, and the initial radius givesHere $[\beta ]=\mathrm{kg}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, $[\rho ]=\mathrm{kg}{\mathrm{m}}^{-3}$, and $[\mu ]=\mathrm{m}{\mathrm{s}}^{-1}$. This surface-fed growth of an accreting sphere law assumes constant droplet mass density.

For surface-fed growth of an accreting sphere with $\mu =\dot{r}>0$, stationary incoming material, and quadratic drag $F_D=C_D{\rho }_a\pi r^2{v}^2/2$, the variable-mass system obeys $\dot{v}+3\mu v/r=g-3C_D{\rho }_a{v}^2/(8\rho r)$. For constant positive parameters, $r\sim \mu t$ and the late balance gives the displayed speed. The omitted acceleration and accretion terms are both $O(t^{-1/2})$, consistent with the balance. Thus this growing sphere has unbounded speed with vanishing acceleration, rather than a finite terminal velocity. This model has a different growth law from the existing accreting raindrop with quadratic drag, whose growth rate is proportional to speed.