If rain falls vertically with a velocity `V_r` and wind blows with a velocity `V_w` from east to west, then a person standing on the roadside should hold the umbrella in the direction

Options

A. `tan theta = V_w/V_r`
B. `tan theta =V_r / V_w`
C. `tan theta =V_(rw)/sqrt(V_r^2+V_w^2)`
D. `tan theta =V_(r)/sqrt(V_r^2+V_w^2)`

Answer

To solve the problem of determining the direction in which a person should hold an umbrella when rain falls vertically and wind blows from east to west, we can follow these steps:

### Step 1: Understand the velocities involved
- The rain falls vertically with a velocity \( V_r \).
- The wind blows horizontally from east to west with a velocity \( V_w \).

### Step 2: Set up a coordinate system
- We can set up a Cartesian coordinate system where:
- The vertical direction (downward) is along the negative y-axis.
- The horizontal direction (from east to west) is along the negative x-axis.

### Step 3: Represent the velocities as vectors
- The velocity of rain can be represented as a vector:
\[ \vec{V_r} = 0\hat{i} - V_r\hat{j} \]
- The velocity of wind can be represented as a vector:
\[ \vec{V_w} = -V_w\hat{i} + 0\hat{j} \]

### Step 4: Find the resultant velocity vector
- The resultant velocity vector \( \vec{V} \) of the rain as perceived by a person on the ground (considering the wind) is the vector sum of \( \vec{V_r} \) and \( \vec{V_w} \):
\[ \vec{V} = \vec{V_r} + \vec{V_w} = -V_w\hat{i} - V_r\hat{j} \]

### Step 5: Determine the angle \( \theta \)
- The angle \( \theta \) that the resultant vector makes with the vertical (y-axis) can be found using the tangent function:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_w}{V_r} \]

### Step 6: Conclusion
- Therefore, the person should hold the umbrella at an angle \( \theta \) such that:
\[ \tan \theta = \frac{V_w}{V_r} \]
- This means the correct option is:
**a. \( \tan \theta = \frac{V_w}{V_r} \)**