Raindrops are falling vertically with a velocity `10 m//s`. To a cyclist moving on a straight road the rain drops appear to be coming with a velocity of `20m//s`. The velocity of cyclist is :-

Options

A. `10m//s`
B. `10sqrt(3)m//s`
C. `20m//s`
D. `20sqrt(3)m//s`

Answer

To find the velocity of the cyclist, we can use the information given about the velocities of the raindrops and the apparent velocity of the rain as seen by the cyclist.

### Step-by-Step Solution:

1. **Identify the velocities**:
- The velocity of the raindrops falling vertically downward, \( V_R = 10 \, \text{m/s} \).
- The apparent velocity of the raindrops as observed by the cyclist, \( V_{RA} = 20 \, \text{m/s} \).

2. **Understand the scenario**:
- The raindrops are falling straight down, and the cyclist is moving horizontally. Therefore, the velocities form a right triangle where:
- One leg is the velocity of the rain \( V_R \) (vertical).
- The other leg is the velocity of the cyclist \( V_C \) (horizontal).
- The hypotenuse is the apparent velocity of the rain \( V_{RA} \).

3. **Apply the Pythagorean theorem**:
- According to the Pythagorean theorem:
\[ V_{RA}^2 = V_R^2 + V_C^2 \]
- Plugging in the known values:
\[ (20 \, \text{m/s})^2 = (10 \, \text{m/s})^2 + V_C^2 \]
- This simplifies to:
\[ 400 = 100 + V_C^2 \]

4. **Solve for \( V_C^2 \)**:
- Rearranging the equation gives:
\[ V_C^2 = 400 - 100 = 300 \]

5. **Calculate \( V_C \)**:
- Taking the square root of both sides:
\[ V_C = \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \, \text{m/s} \]

6. **Final answer**:
- The velocity of the cyclist is \( V_C = 10\sqrt{3} \, \text{m/s} \).

### Summary:
The velocity of the cyclist is \( 10\sqrt{3} \, \text{m/s} \).