A car with a vertical windshield moves in a rain storm at a speed of 40 km/hr. The rain drops fall vertically with constant speed of 20 m/s. The angle at which rain drops strike the windshield is -

Options

A. ` tan^(-1) (5)/(9)`
B. `tan^(-1) (9)/(5)`
C. `tan^(-1) (3)/(2)`
D. `tan^(-1) (2)/(3)`

Answer

To solve the problem of finding the angle at which raindrops strike the windshield of a car moving in a rainstorm, we can follow these steps:

### Step 1: Convert the speed of the car from km/hr to m/s
The speed of the car is given as 40 km/hr. To convert this to meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{1000 \text{ m}}{3600 \text{ s}} \]
Calculating this:
\[ 40 \text{ km/hr} = 40 \times \frac{1000}{3600} = \frac{40000}{3600} \approx 11.11 \text{ m/s} \]

### Step 2: Identify the velocities
- The velocity of the car \( V_c \) is approximately \( 11.11 \text{ m/s} \) (horizontal direction).
- The velocity of the raindrops \( V_r \) is \( 20 \text{ m/s} \) (vertical direction).

### Step 3: Determine the relative velocity of the rain with respect to the car
Since the car is moving forward, the raindrops will appear to come at an angle. The horizontal component of the relative velocity is the speed of the car, and the vertical component is the speed of the raindrops.

### Step 4: Use the tangent function to find the angle
The angle \( \theta \) at which the raindrops strike the windshield can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{Vertical component}}{\text{Horizontal component}} = \frac{V_r}{V_c} \]
Substituting the values we have:
\[ \tan(\theta) = \frac{20 \text{ m/s}}{11.11 \text{ m/s}} \approx 1.8 \]

### Step 5: Calculate the angle
To find \( \theta \), we take the arctangent (inverse tangent) of the result:
\[ \theta = \tan^{-1}(1.8) \]
Using a calculator, we find:
\[ \theta \approx 60.95^\circ \]

### Conclusion
The angle at which the raindrops strike the windshield is approximately \( 60.95^\circ \).

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