$\mathrm{restart}\:$

$\mathrm{Fd}≔\frac{1}{2}\cdot \mathrm{\π}\cdot \frac{{\mathrm{Dia}}^2}{4}\cdot \mathrm{CD}\cdot \mathrm{\ρ\_\_f}\cdot v^2\:$

$\mathrm{Fb}≔\left(\mathrm{\ρ\_\_p}-\mathrm{\ρ\_\_f}\right)\cdot g\cdot \mathrm{\π}\cdot \frac{{\mathrm{Dia}}^3}{6}\:$

$\mathrm{res}≔\mathrm{v\_\_terminal}\=\mathrm{solve}\left(\mathrm{Fb}\=\mathrm{Fd}\,v\right)\left[1\right]$

$\mathrm{res}≔\mathrm{v\_\_terminal}=\frac{2\sqrt{-3\mathrm{CD}\mathrm{\ρ\_\_f}g\mathrm{Dia}\left(\mathrm{\ρ\_\_f}-\mathrm{\ρ\_\_p}\right)}}{3\mathrm{CD}\mathrm{\ρ\_\_f}}$

$g≔9.81\:$

$\mathrm{\ρ\_\_p}≔1800\:$

$\mathrm{\ρ\_\_f}≔994.6\:$

$\mathrm{\μ}≔0.0008931\:$

$\mathrm{Dia}≔0.000208\:$

$\mathrm{CD}≔\mathrm{Rey}\to \&lcub;\begin{array}{cc}\frac{24}{\mathrm{Rey}}& \mathrm{Rey}<0.1\\ \frac{24}{\mathrm{Rey}}\left(1\&plus;0.14\cdot {\mathrm{Rey}}^{0.7}\right)& 0.1\le \mathrm{Rey}\le 1000\\ 0.44& 1000\le \mathrm{Rey}\le 350000\\ 0.19-\frac{8\cdot {10}^4}{\mathrm{Rey}}& \mathrm{Rey}\&gt;350000\end{array}\&colon;$

$\mathrm{ReynoldsNumber}≔\mathrm{Rey}\&equals;\frac{\mathrm{Dia} \mathrm{\&rho;\_\_f} \mathrm{v\_\_terminal}}{\mathrm{\&mu;}}\&colon;$

$\mathrm{TerminalVelocity}≔\mathrm{v\_\_terminal}\&equals;\sqrt{\frac{4g\left(\mathrm{\&rho;\_\_p}-\mathrm{\&rho;\_\_f}\right)\mathrm{Dia}}{3\mathrm{CD}\left(\mathrm{Rey}\right)\cdot \mathrm{\&rho;\_\_f}}}\&colon;$

$\mathrm{fsolve}\left(\left\{\mathrm{ReynoldsNumber}\&comma;\mathrm{TerminalVelocity}\right\}\&comma;\left\{\mathrm{v\_\_terminal}\&comma;\mathrm{Rey}\right\}\right)\&semi;$$$

$\left\{\mathrm{Rey}=3.656696458\&comma;\mathrm{v\_\_terminal}=0.01578618582\right\}$