Two rain drops reach the earth with their terminal velocities in the ratio 4 : 9. Find the ratio of their radii .

Explanation

To solve the problem of finding the ratio of the radii of two raindrops given their terminal velocities in the ratio of 4:9, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Terminal Velocity and Radius**:
The terminal velocity \( V \) of a spherical object falling through a fluid is given by the equation:
\[ V = \frac{2}{9} \frac{g (r^2)(\rho - \sigma)}{\eta} \]
where:
- \( g \) is the acceleration due to gravity,
- \( r \) is the radius of the drop,
- \( \rho \) is the density of the raindrop,
- \( \sigma \) is the density of the surrounding fluid,
- \( \eta \) is the coefficient of viscosity of the fluid.

2. **Establish the Given Ratio of Terminal Velocities**:
We are given that the terminal velocities \( V_1 \) and \( V_2 \) are in the ratio:
\[ \frac{V_1}{V_2} = \frac{4}{9} \]

3. **Relate the Radii to Terminal Velocities**:
Since the terminal velocity is proportional to the square of the radius, we can express this as:
\[ V \propto r^2 \]
Thus, we can write:
\[ \frac{V_1}{V_2} = \frac{r_1^2}{r_2^2} \]

4. **Substitute the Given Ratio into the Equation**:
From the ratio of terminal velocities:
\[ \frac{4}{9} = \frac{r_1^2}{r_2^2} \]

5. **Take the Square Root to Find the Ratio of Radii**:
Taking the square root of both sides gives:
\[ \frac{r_1}{r_2} = \sqrt{\frac{4}{9}} = \frac{2}{3} \]

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 their radii is \( \frac{2}{3} \).

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