Rain is falling vertically with a velocity of `25 ms^-1`. A woman rides a bicycle with a speed of `10 ms^-1` in the north to south direction. What is the direction (angle with vertical) in which she should hold her umbrella to safe herself from rain ?

Options

A. a. `tan^-1 (0.4)`
B. b. `tan^-1 (1)`
C. c. `tan^-1 (sqrt(3))`
D. d. `tan^-1 (2.6)`

Explanation

To solve the problem of determining the angle at which the woman should hold her umbrella to protect herself from the rain, we can follow these steps:

### Step 1: Understand the velocities involved
- The rain is falling vertically downwards with a velocity \( V_R = 25 \, \text{m/s} \).
- The woman is riding a bicycle from north to south with a velocity \( V_W = 10 \, \text{m/s} \).

### Step 2: Set up the relative velocity
To find out how the rain appears to the woman, we need to consider the relative velocity of the rain with respect to the woman. The relative velocity \( V_{RW} \) can be expressed as:
\[ V_{RW} = V_R - V_W \]
Since the rain is falling vertically downwards and the woman is moving horizontally, we can represent this as a right triangle where:
- The vertical component is the velocity of the rain \( V_R = 25 \, \text{m/s} \).
- The horizontal component is the velocity of the woman \( V_W = 10 \, \text{m/s} \).

### Step 3: Calculate the angle
To find the angle \( \theta \) with respect to the vertical, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{horizontal component}}{\text{vertical component}} = \frac{V_W}{V_R} \]
Substituting the values:
\[ \tan(\theta) = \frac{10 \, \text{m/s}}{25 \, \text{m/s}} = 0.4 \]

### Step 4: Find the angle using the inverse tangent function
Now, we can find \( \theta \) by taking the inverse tangent:
\[ \theta = \tan^{-1}(0.4) \]

### Step 5: Calculate the angle
Using a calculator, we find:
\[ \theta \approx 21.8^\circ \]

### Conclusion
The woman should hold her umbrella at an angle of approximately \( 21.8^\circ \) with respect to the vertical to protect herself from the rain.

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