Question

Two raindrops reach the earth with different terminal velocities having the ratio \(9: 4\). The ratio of their volumes is

• A. \(\frac{8}{27}\)
• B. \(\frac{9}{4}\)
• C. \(\frac{3}{2}\)
• D. \(\frac{27}{8}\)

Hint:

Terminal velocity of a falling drop is proportional to the square of its radius, and volume is proportional to the cube of the radius.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For a small drop falling in air, the terminal velocity is \(v_t \propto r^2\).
Volume of a spherical drop is \(V \propto r^3\).

Step 2: Find the ratio of radii
\(\dfrac{v_1}{v_2}=\dfrac{r_1^2}{r_2^2}=\dfrac{9}{4}\)
So \(\dfrac{r_1}{r_2}=\dfrac{3}{2}\).

Step 3: Find the ratio of volumes
\(\dfrac{V_1}{V_2}=\left(\dfrac{r_1}{r_2}\right)^3=\left(\dfrac{3}{2}\right)^3=\dfrac{27}{8}\)

Step 4: Compare the options
The ratio \(\dfrac{27}{8}\) matches the last option.

Final Answer:
The ratio of the volumes of the two raindrops is 27 : 8. \[ \boxed{\text{(D) }\dfrac{27}{8}} \]