The ratio of the terminal velocities of two drops of radii R and `R//2` is

Options

A. `2`
B. `1`
C. `1//2`
D. `4`

Explanation

To find the ratio of the terminal velocities of two drops with radii \( R \) and \( \frac{R}{2} \), we can use the relationship between terminal velocity and radius.

### Step-by-Step Solution:

1. **Understand the relationship**: The terminal velocity \( v \) of a spherical object falling through a fluid is given by the equation:
\[ v \propto r^2 \]
where \( r \) is the radius of the sphere.

2. **Define the radii**: Let the radii of the two drops be:
- \( r_1 = R \)
- \( r_2 = \frac{R}{2} \)

3. **Express terminal velocities**: Let the terminal velocities corresponding to these radii be \( v_1 \) and \( v_2 \). According to the relationship:
\[ v_1 \propto r_1^2 \quad \text{and} \quad v_2 \propto r_2^2 \]

4. **Write the proportionality**: We can express the terminal velocities as:
\[ v_1 = k \cdot r_1^2 \quad \text{and} \quad v_2 = k \cdot r_2^2 \]
where \( k \) is a constant of proportionality.

5. **Substitute the radii**:
- For \( v_1 \):
\[ v_1 = k \cdot R^2 \]
- For \( v_2 \):
\[ v_2 = k \cdot \left(\frac{R}{2}\right)^2 = k \cdot \frac{R^2}{4} \]

6. **Find the ratio of terminal velocities**:
\[ \frac{v_1}{v_2} = \frac{k \cdot R^2}{k \cdot \frac{R^2}{4}} = \frac{R^2}{\frac{R^2}{4}} = \frac{R^2 \cdot 4}{R^2} = 4 \]

7. **Conclusion**: The ratio of the terminal velocities of the two drops is:
\[ \frac{v_1}{v_2} = 4 \]

### Final Answer:
The ratio of the terminal velocities of the two drops is \( 4 \).

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