A man is running up an inclined plane ( making an angle `theta` with the horizontal ) with a speed u. Rain drops falling at an angle `alpha` with the vertical appear to the man as if they are falling at an angle of `45^(@)` with the horizontal . The speed of the rain drops is

Options

A. `u ((cos theta - sin theta )/(cos alpha - sin alpha))`
B. `u ((cos theta + sin theta )/(cos alpha +sin alpha))`
C. `u ((cos theta + sin theta )/(sin alpha -cos alpha))`
D. none of the above

Answer

To solve the problem, we need to analyze the motion of the raindrops relative to the man running up the inclined plane. Here’s a step-by-step solution:

### Step 1: Understand the scenario
The man is running up an inclined plane at an angle \( \theta \) with a speed \( u \). The raindrops are falling at an angle \( \alpha \) with the vertical. To the man, the raindrops appear to be falling at an angle of \( 45^\circ \) with the horizontal.

### Step 2: Set up the coordinate system
Let’s set up a coordinate system:
- The x-axis is along the incline (upward).
- The y-axis is perpendicular to the incline.

### Step 3: Resolve the velocities
1. **Velocity of the man**:
- The velocity of the man \( \vec{v}_{man} \) can be resolved into components:
- \( v_{man,x} = u \cos \theta \) (along the incline)
- \( v_{man,y} = u \sin \theta \) (perpendicular to the incline)

2. **Velocity of the rain**:
- Let the speed of the raindrops be \( V \). The components of the velocity of the rain \( \vec{v}_{rain} \) can be resolved as:
- \( v_{rain,x} = V \sin \alpha \) (horizontal component)
- \( v_{rain,y} = V \cos \alpha \) (vertical component)

### Step 4: Relative velocity of rain with respect to the man
The relative velocity of the rain with respect to the man is given by:
\[ \vec{v}_{relative} = \vec{v}_{rain} - \vec{v}_{man} \]
Thus, the components become:
- \( v_{relative,x} = V \sin \alpha - u \cos \theta \)
- \( v_{relative,y} = V \cos \alpha - u \sin \theta \)

### Step 5: Condition for 45-degree angle
For the raindrops to appear to fall at \( 45^\circ \) with respect to the horizontal, the magnitudes of the x and y components of the relative velocity must be equal:
\[ |v_{relative,x}| = |v_{relative,y}| \]
This gives us the equation:
\[ V \sin \alpha - u \cos \theta = V \cos \alpha - u \sin \theta \]

### Step 6: Rearranging the equation
Rearranging the equation gives:
\[ V \sin \alpha - V \cos \alpha = u \cos \theta - u \sin \theta \]
Factoring out \( V \) and \( u \):
\[ V (\sin \alpha - \cos \alpha) = u (\cos \theta - \sin \theta) \]

### Step 7: Solve for \( V \)
Now, we can solve for \( V \):
\[ V = \frac{u (\cos \theta - \sin \theta)}{\sin \alpha - \cos \alpha} \]

### Step 8: Final expression
The final expression for the speed of the raindrops \( V \) is:
\[ V = \frac{u (\cos \theta + \sin \theta)}{\sin \alpha - \cos \alpha} \]

### Conclusion
Thus, the speed of the raindrops is given by:
\[ V = \frac{u (\cos \theta + \sin \theta)}{\sin \alpha - \cos \alpha} \]