Two rain drop reach the earth with their with the terminal velocity in the ratio 4 :9 . The ratio of their radii is

Options

A. `4:09`
B. `2:3`
C. `3:2`
D. `9:4`

Answer

To solve the problem of finding the ratio of the radii of two raindrops given their terminal velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Terminal Velocity**:
The terminal velocity (V_t) of a spherical object falling through a fluid is given by the formula:
\[ V_t = \frac{2}{9} \cdot \frac{(ρ_p - ρ_l) \cdot g \cdot r^2}{η} \]
where:
- \(ρ_p\) = density of the particle (raindrop)
- \(ρ_l\) = density of the liquid (air in this case)
- \(g\) = acceleration due to gravity
- \(r\) = radius of the spherical object
- \(η\) = coefficient of viscosity of the fluid

2. **Given Information**:
We are given that the ratio of the terminal velocities of the two raindrops is:
\[ \frac{V_{t1}}{V_{t2}} = \frac{4}{9} \]

3. **Setting Up the Ratio**:
Since the terminal velocity is proportional to the square of the radius (assuming other factors remain constant), we can write:
\[ \frac{V_{t1}}{V_{t2}} = \frac{r_1^2}{r_2^2} \]
where \(r_1\) and \(r_2\) are the radii of the first and second raindrop, respectively.

4. **Substituting the Given Ratio**:
Plugging in the given ratio of terminal velocities:
\[ \frac{4}{9} = \frac{r_1^2}{r_2^2} \]

5. **Finding the Ratio of Radii**:
To find the ratio of the radii, we take the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

6. **Conclusion**:
Therefore, the ratio of the radii of the two raindrops is:
\[ \frac{r_1}{r_2} = \frac{2}{3} \]

### Final Answer:
The ratio of the radii of the two raindrops is \( \frac{2}{3} \).

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