Accreting raindrop with quadratic drag ($\dot{a}=cv/4,\dot{v}=g-3(c+k)v^2/(4a)$)

For a spherical drop of constant mass density $\rho$, radius $a>0$ and downward velocity $v>0$, suppose its volume grows at rate $c\pi a^{2}v$ by collecting stationary mist and its upward quadratic drag is $k\rho \pi a^2{v}^2$, with $c,k>0$. The drop mass is $4\pi \rho a^3/3$. Its incoming mass accretion rate and the variable-mass system momentum law give the displayed equations. The accreted matter must be accelerated from rest, producing the additional $c$ term alongside drag.

For the accreting raindrop with quadratic drag, set $q=v^2/a$ and $B=7c/8+3k/4$. Its exact equation is $\dot{q}=2(v/a)(g-Bq)$. Dividing by $\dot{a}=cv/4$ gives $q(a)=g/B+(q(a_0)-g/B)(a_0/a{)}^{7+6k/c}$. Thus $q$ stays between its positive initial value and positive equilibrium. Since $d\sqrt{a}/dt=c\sqrt{q}/8$, the radius grows without finite-time divergence and tends to infinity. The power correction then vanishes, proving both limits. Positive drag makes the limiting acceleration strictly less than $g/7$.