Rain is falling vertically with a speed of `35 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 ?

Answer

To find the direction in which the woman should hold her umbrella while riding her bicycle in the rain, we can analyze the problem step by step.

### Step 1: Understand the velocities involved
- The rain is falling vertically downward with a speed of \( V_r = 35 \, \text{m/s} \).
- The woman is riding her bicycle from east to west with a speed of \( V_w = 12 \, \text{m/s} \).

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

### Step 3: Determine the velocities in vector form
- The velocity of the rain can be represented as:
\[ \vec{V_r} = 0 \hat{i} - 35 \hat{j} \, \text{(downward)} \]
- The velocity of the woman can be represented as:
\[ \vec{V_w} = -12 \hat{i} + 0 \hat{j} \, \text{(westward)} \]

### Step 4: Calculate the relative velocity of the rain with respect to the woman
- The relative velocity of the rain with respect to the woman is given by:
\[ \vec{V_{rw}} = \vec{V_r} - \vec{V_w} \]
- Substituting the values:
\[ \vec{V_{rw}} = (0 \hat{i} - 35 \hat{j}) - (-12 \hat{i} + 0 \hat{j}) = 12 \hat{i} - 35 \hat{j} \]

### Step 5: Determine the angle at which the umbrella should be held
- The direction in which she should hold her umbrella is opposite to the direction of \( \vec{V_{rw}} \).
- To find the angle \( \theta \) with respect to the vertical, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{35} \]
- Therefore, we can find \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{12}{35}\right) \]

### Step 6: Calculate the angle
- Using a calculator:
\[ \theta \approx \tan^{-1}(0.342857) \approx 18.92^\circ \]

### Step 7: Direction of the umbrella
- Since the rain is falling vertically and the woman is moving westward, she should hold her umbrella at an angle of approximately \( 18.92^\circ \) from the vertical towards the east.

### Final Answer
The woman should hold her umbrella at an angle of approximately \( 18.92^\circ \) towards the east from the vertical.

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