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 determine 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 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 establish a coordinate system where:
- The downward direction (rain) is negative y-axis.
- The east to west direction (woman's motion) is along the negative x-axis.

### Step 3: Represent the velocities as vectors
- The velocity of the rain can be represented as:
\[ \vec{v_r} = (0, -35) \, \text{m/s} \]
- The velocity of the woman can be represented as:
\[ \vec{v_w} = (-12, 0) \, \text{m/s} \]

### Step 4: Calculate the relative velocity of 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:
\[ \vec{v_{rw}} = (0, -35) - (-12, 0) = (12, -35) \, \text{m/s} \]

### Step 5: Determine the direction of the umbrella
- To find the angle \( \theta \) at which the umbrella should be held, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{v_r}{v_w} = \frac{35}{12} \]
- Therefore,
\[ \theta = \tan^{-1}\left(\frac{35}{12}\right) \]

### Step 6: Calculate the angle
- Using a calculator:
\[ \theta \approx \tan^{-1}(2.9167) \approx 71.57^\circ \]

### Step 7: Conclusion
- The woman should hold her umbrella at an angle of approximately \( 71.57^\circ \) from the horizontal (towards the direction of the rain) to shield herself from the rain.