Rain is falling vertically with a speed of `30 m s^(-1)` . A woman rides a bicycle with a speed of ` 12 m s^(-1)` in east to west direction. In which direction should she hold her umbrella ?

Options

A. At an angle of `tan^(-1)((2)/(5))` with the vertical towards the east.
B. At angle of `tan^(-1)((2)/(5))` with the vertical towards the west.
C. At angle of `tan^(-1)((5)/(2))` with the vertical towards the west
D. At angle of `tan^(-1)((5)/(2))` with the vertical towards the west.

Explanation

To solve the problem of determining the direction in which the woman should hold her umbrella while riding her bicycle in the rain, we can follow these steps:

### Step 1: Identify the velocities
- The rain is falling vertically downwards with a speed of \( V_r = 30 \, \text{m/s} \).
- The woman is riding her bicycle from east to west with a speed of \( V_b = 12 \, \text{m/s} \).

### Step 2: Establish the coordinate system
- Let's define the downward direction (where the rain is falling) as the negative y-direction.
- The east-west direction can be defined as the x-axis, where east is positive and west is negative.

### Step 3: Determine the relative velocity of rain with respect to the bicycle
- The velocity of the rain can be represented as a vector: \( \vec{V_r} = (0, -30) \, \text{m/s} \) (0 in the x-direction and -30 in the y-direction).
- The velocity of the bicycle can be represented as: \( \vec{V_b} = (-12, 0) \, \text{m/s} \) (moving west).

### Step 4: Calculate the relative velocity
- The relative velocity of the rain with respect to the bicycle is given by:
\[ \vec{V_{r/b}} = \vec{V_r} - \vec{V_b} = (0, -30) - (-12, 0) = (12, -30) \, \text{m/s} \]
- This means that the rain appears to be coming from a direction with a horizontal component of \( 12 \, \text{m/s} \) (to the east) and a vertical component of \( 30 \, \text{m/s} \) (downwards).

### Step 5: Determine the angle of the umbrella
- To find the angle \( \theta \) at which she should hold her umbrella, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{V_b}{V_r} = \frac{12}{30} = \frac{2}{5} \]
- Therefore, the angle \( \theta \) is:
\[ \theta = \tan^{-1}\left(\frac{2}{5}\right) \]

### Step 6: Determine the direction
- The umbrella should be held in the direction of the relative velocity of the rain with respect to the bicycle, which is towards the east and downward. Thus, the woman should hold her umbrella at an angle \( \theta \) from the vertical towards the west.

### Final Answer
The woman should hold her umbrella at an angle of \( \tan^{-1}\left(\frac{2}{5}\right) \) from the vertical towards the west.

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