If n identical water droplets falling under gravity with terminal velocity v coalesce to form a single drop which has the terminal velocity $4$v, find the number n.

No of identical water dropletts falling falling under gravity$=n$

Thermal velocity$=V$

And then coalesce to form a single drop

The thermal velocity $==4V$

Find n

The terminal velocity of an object is directly proportional to the square of the radius of the drop. The volume of the small drop is $\frac{4}{3}\pi r^3$. The volume of the bigger drop formed by collection of  $N$ drops is $N\frac{4}{3}\pi R^3$

(R be the radius of the bigger droplet and r be the radius of the smaller droplet)

Thus the radius of the bigger drop is $$\begin{gathered} R=N^{1/3}\times r(\frac{V_2}{V_1}) \\ =(\frac{R}{r}{)}^2)(\frac{V_2}{V_1}) \\ =(\frac{r-N^{1/3}}{r}{)}^2\times \frac{V_2}{V_1} \\ =N^{2/3}4V \\ =N^{2/3}V \\ N=8 \end{gathered}$$