A shower of rain appears to fall vertically downwards with a velocity of 12 kmph on a person walking west wards with a velocity of 5 kmph. The actual velocity and direction of the rain are

Options

A. 7.5 kmph, clockwise to vertical
B. 13 kmph, anti clockwise to vertical
C. 13 kmph, clockwise to vertical
D. 17 kmph, clockwise to vertical

Answer

To solve the problem, we need to find the actual velocity and direction of the rain, given that it appears to fall vertically downwards to a person walking westward.

### Step-by-Step Solution:

1. **Identify the velocities:**
- Let \( V_m \) be the velocity of the man (person walking westward) = 5 km/h.
- Let \( V_{rm} \) be the velocity of the rain with respect to the man = 12 km/h (appears vertical).

2. **Set up the coordinate system:**
- Assume the positive x-direction is east and the positive y-direction is upwards.
- The man's velocity \( V_m \) is towards the west, which can be represented as \( V_m = -5 \hat{i} \) km/h.
- The rain's velocity with respect to the man \( V_{rm} \) is vertical downwards, represented as \( V_{rm} = -12 \hat{j} \) km/h.

3. **Calculate the actual velocity of the rain:**
- The actual velocity of the rain \( V_r \) can be found using vector addition:
\[ V_r = V_{rm} + V_m \]
- Since the rain appears to fall vertically downwards, we can represent the actual velocity of the rain as:
\[ V_r = V_{rm} \hat{j} + V_m \hat{i} \]
- This gives us:
\[ V_r = -12 \hat{j} - 5 \hat{i} \]

4. **Find the magnitude of the resultant velocity:**
- The magnitude of the resultant velocity \( |V_r| \) is given by:
\[ |V_r| = \sqrt{(-5)^2 + (-12)^2} \]
- Calculating this:
\[ |V_r| = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ km/h} \]

5. **Determine the direction of the rain:**
- The direction can be found using the tangent of the angle \( \theta \) with respect to the vertical:
\[ \tan(\theta) = \frac{\text{horizontal component}}{\text{vertical component}} = \frac{5}{12} \]
- This angle is measured clockwise from the vertical.

6. **Conclusion:**
- The actual velocity of the rain is 13 km/h and it is directed clockwise to the vertical.

### Final Answer:
The actual velocity of the rain is **13 km/h** in the **clockwise direction to the vertical**.