$$\begin{aligned} & \mathrm{mg}-\mathrm{F}_{\mathrm{B}}-\mathrm{F}_{\mathrm{v}}=\mathrm{ma} \\ & \left(\rho \frac{4}{3} \pi \mathrm{r}^3\right) g-\left(\rho_{\mathrm{L}} \frac{4}{3} \pi \mathrm{r}^3\right) g-6 \pi \eta r v=m \frac{d v}{d t} \end{aligned}$$

Let $$\frac{4}{3 m} \pi R^3 g\left(\rho-\rho_L\right)=K_1$$ and $$\frac{6 \pi \eta r}{m}=K_2$$

$$\frac{\mathrm{dv}}{\mathrm{dt}}=\mathrm{K}_1-\mathrm{K}_2 \mathrm{~V}$$

$$\begin{aligned} & \int_\limits0^v \frac{d v}{K_1-K_2 v}=\int_\limits0^t d t \\ & -\frac{1}{K_2} \ln \left[K_1-K_2 v\right]_0^v=t \\ & \ell n\left(\frac{K_1-K_2 v}{K_1}\right)=-K_2 t \\ & K_1-K_2 v=K_1 e^{-K_2 t} \\ & v=\frac{K_1}{K_2}\left[1-e^{-K_2 t}\right] \end{aligned}$$