= Solution
For an initially monodisperse cloud the relative settling speed vanishes, so there is no gravitational collection term in this model. Set $D=(S-1)/(F_k+F_d)$ and use $M=4\pi\rho_l r^3/3$. Condensational growth gives
$$
4\pi\rho_l r^2\dot r=4\pi\rho_l rD,\qquad r^2(t)=r_0^2+2Dt.
$$
Taking upward height from cloud base as positive, the droplet speed relative to the ground is the updraft minus its downward <terminal velocity>:
$$
\dot h_r=U-k_rr^2(t)=U-k_r(r_0^2+2Dt).
$$
For a launch at $h_r(0)=0$,
$$
\boxed{h_r(t)=(U-k_rr_0^2)t-k_rDt^2.}
$$
This assumes the prescribed uniform updraft and supersaturation persist along the trajectory, and that the Stokes-type radius-squared fall law remains applicable.
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