 Solution

Neglect condensation and the smaller droplets' fall speed, while retaining their specified liquid-water content. For exponential cloud collector growth, the collection law and spherical collector mass giveThus $R(t)=R_0{e}^{{\alpha }_{R}t}$. The growing terminal speed eventually exceeds the updraft, andwhere the last expression uses $t$ in seconds and $U=1\mathrm{m}{\mathrm{s}}^{-1}$. Numerically,The trajectory reaches maximum height when $k_{R}R=U$, at $t=1145\mathrm{s}$, $R=125\mu \mathrm{m}$ and $h_{\max }=620\mathrm{m}$. It returns to cloud base near $t=1860\mathrm{s}$, with radius about $392\mu \mathrm{m}$. The left-to-right sketch in the preceding figure shows this rise and subsequent fall.

Once the drop leaves the cloud, the prescribed in-cloud $L$, $U$ and collection law need not persist. Consequently the negative $2000$-second value describes the requested constant-coefficient model's extension, not a prediction that collection continues below cloud base. Its actual subsequent height needs an outside-cloud model. Within the cloud, the exponential feedback between increasing radius and increasing swept volume explains how a minority of collector drops can escape the condensation bottleneck and initiate warm rain; depletion, breakup and changes in fall law eventually limit indefinite exponential growth.