Question

125 small water drops of same size fall through air with constant velocity 4 cm/s. They coalesce to form a big drop. The terminal velocity of the big drop is ______.

• A. 0.5 m/s
• B. 1 m/s
• C. 1.5 m/s
• D. 2.5 m/s

Hint:

Shortcut for coalescing drops: The new terminal velocity is always $V_{new} = V_{old} \times (N^{2/3})$. Here, $N=125$, so $125^{2/3} = (5^3)^{2/3} = 5^2 = 25$. Multiply old speed by 25!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Multiple small fluid drops falling at terminal velocity merge into a single large drop. We must find the terminal velocity of the newly formed large drop.

Step 2: Detailed Explanation:
The terminal velocity ($v_t$) of a spherical drop falling through a viscous fluid (like air) is given by Stokes' Law:
$v_t = \frac{2}{9} \frac{r^2 (\rho - \sigma) g}{\eta}$
where $r$ is the radius of the drop.
Since all other fluid parameters remain constant, the terminal velocity is strictly proportional to the square of the radius:
$v_t \propto r^2$
1. Find the radius of the big drop ($R$) relative to the small drop ($r$):
When $N$ small drops coalesce, their total physical volume is perfectly conserved.
$\text{Volume of Big Drop} = N \times \text{Volume of Small Drop}$
$\frac{4}{3} \pi R^3 = 125 \times \left(\frac{4}{3} \pi r^3\right)$
$R^3 = 125 r^3$
Take the cube root of both sides:
$R = \sqrt[3]{125} r = 5r$
2. Relate the terminal velocities:
Let $V_T$ be the terminal velocity of the big drop, and $v_t$ be the terminal velocity of the small drop.
Using the proportionality $v_t \propto r^2$, we can set up a ratio:
$\frac{V_T}{v_t} = \left(\frac{R}{r}\right)^2$
Substitute $R = 5r$:
$\frac{V_T}{v_t} = \left(\frac{5r}{r}\right)^2 = 5^2 = 25$
This means the big drop falls 25 times faster than the small drops!
3. Calculate final velocity:
We are given $v_t = 4 \text{ cm/s}$.
$V_T = 25 \times 4 \text{ cm/s} = 100 \text{ cm/s}$.
Convert centimeters to meters to match the options:
$100 \text{ cm/s} = 1 \text{ m/s}$.

Step 3: Final Answer:
The terminal velocity of the big drop is 1 m/s, matching option (b).