Question

Eight equal drops of water are falling through air with a steady velocity of $5\text{cm/sec}$. If smaller drops combine to form a single large drop, then the terminal velocity (in $\text{cm/sec}$) of this large drop is

• A. $5$
• B. $20$
• C. $10$
• D. $15$

Answer 1

The correct option is B $20$

Let the terminal velocity of larger drop $=V_{\text{T large}}$
Let the radius of the small drop $=r$
Radius of the bigger drop $=R$
Given, terminal velocity of small drop $V_{\text{T small}}$ $=5\text{cm/s}$
When $8$ such drops combine to form a bigger drop the total volume of water remains same.
So, $8\times \frac{4}{3}\pi r^3=\frac{4}{3}\pi R^3$
$\Rightarrow R=2r$
Terminal velocity $\left(V_T\right)$ is given by $V_T=\frac{2(P_{water}-P_{air})r^{2}g}{9\mu }$
$\Rightarrow V_T\propto r^2$
$\Rightarrow \frac{V_{\text{T small}}}{V_{\text{T large}}}=\frac{r^2}{R^2}[\because R=2r]$
$\Rightarrow \frac{5}{V_{\text{T large}}}=\frac{r^2}{4r^2}\Rightarrow V_{\text{T large}}=20\text{cm/sec}$