A man starts running along a straight road with uniform velocity observes that the rain is falling vertically downward. If he doubles his speed, he finds that the rain is coming at an angle `theta` to the vertical. The velocity of rain with respect to the ground is :

Options

A. `uhati-utanthetahatj`
B. `uhati-ucotthetahatj`
C. `uhati+ucotthetahatj`
D. `(u)/(tantheta)hati-uhatj`

Explanation

To solve the problem, we will analyze the motion of the man and the rain using vector components.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the velocity of the man be \( u \) (in the horizontal direction).
- Let the velocity of the rain with respect to the ground be \( \vec{V_r} = x \hat{i} + y \hat{j} \), where \( x \) is the horizontal component and \( y \) is the vertical component.

2. **First Observation**:
- The man observes that the rain is falling vertically downward when he runs with velocity \( u \). This means that the horizontal component of the rain's velocity must equal the man's velocity for him to perceive the rain as vertical.
- Therefore, we have:
\[ x - u = 0 \implies x = u \]

3. **Second Observation**:
- When the man doubles his speed to \( 2u \), he observes that the rain is coming at an angle \( \theta \) to the vertical.
- The velocity of the rain with respect to the man is given by:
\[ \vec{V_{rm}} = \vec{V_r} - \vec{V_m} = (x \hat{i} + y \hat{j}) - (2u \hat{i}) = (x - 2u) \hat{i} + y \hat{j} \]

4. **Substituting the Value of \( x \)**:
- From the first observation, we know \( x = u \). Substituting this into the equation gives:
\[ \vec{V_{rm}} = (u - 2u) \hat{i} + y \hat{j} = -u \hat{i} + y \hat{j} \]

5. **Finding the Angle**:
- The rain makes an angle \( \theta \) with the vertical. Therefore, we can relate the components using the tangent function:
\[ \tan \theta = \frac{-u}{y} \]
- Rearranging gives:
\[ y = -\frac{u}{\tan \theta} = -u \cot \theta \]

6. **Final Velocity of Rain**:
- Now we have both components of the rain's velocity:
- \( x = u \)
- \( y = -u \cot \theta \)
- Thus, the velocity of the rain with respect to the ground is:
\[ \vec{V_r} = u \hat{i} - u \cot \theta \hat{j} \]

### Conclusion:
The velocity of the rain with respect to the ground is:
\[ \vec{V_r} = u \hat{i} - u \cot \theta \hat{j} \]