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. `10sqrt3m//s`
C. `20m//s`
D. `20sqrt3m//s`

Explanation

To solve the problem, we will use the concept of relative velocity and the Pythagorean theorem.

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The velocity of the raindrops falling vertically (V_r) = 10 m/s (downward).
- The apparent velocity of the raindrops as observed by the cyclist (V_a) = 20 m/s.

2. **Visualize the Situation**:
- The raindrops are falling straight down, and the cyclist is moving horizontally.
- We can represent the velocities as vectors: the vertical velocity of the rain and the horizontal velocity of the cyclist.

3. **Set Up the Relationship**:
- The apparent velocity of the rain (V_a) can be considered as the resultant of the vertical velocity of the rain (V_r) and the horizontal velocity of the cyclist (V_c).
- According to the Pythagorean theorem:
\[ V_a^2 = V_r^2 + V_c^2 \]

4. **Substitute the Known Values**:
- Substitute V_a = 20 m/s and V_r = 10 m/s into the equation:
\[ (20)^2 = (10)^2 + V_c^2 \]
- This simplifies to:
\[ 400 = 100 + V_c^2 \]

5. **Solve for V_c**:
- Rearranging gives:
\[ V_c^2 = 400 - 100 \]
\[ V_c^2 = 300 \]
- Taking the square root:
\[ V_c = \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \text{ m/s} \]

6. **Conclusion**:
- The velocity of the cyclist is \( 10\sqrt{3} \) m/s.

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

---