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

### Step 1: Identify the velocities
- The car is moving at a speed of 40 km/hr.
- The raindrops are falling vertically at a speed of 20 m/s.

### Step 2: Convert the car's speed to m/s
To convert the speed of the car from km/hr to m/s, we use the conversion factor:
\[ 1 \text{ km/hr} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ 40 \text{ km/hr} = 40 \times \frac{1}{3.6} \text{ m/s} \approx 11.11 \text{ m/s} \]

### Step 3: Set up the velocity vectors
- The velocity of the rain (falling vertically) can be represented as:
\[ \vec{V}_{rain} = 0 \hat{i} - 20 \hat{j} \text{ m/s} \]
- The velocity of the car (moving horizontally) can be represented as:
\[ \vec{V}_{car} = 11.11 \hat{i} + 0 \hat{j} \text{ m/s} \]

### Step 4: Find the relative velocity of rain with respect to the car
The relative velocity of the rain with respect to the car is given by:
\[ \vec{V}_{relative} = \vec{V}_{rain} - \vec{V}_{car} \]
Substituting the vectors:
\[ \vec{V}_{relative} = (0 \hat{i} - 20 \hat{j}) - (11.11 \hat{i} + 0 \hat{j}) = -11.11 \hat{i} - 20 \hat{j} \text{ m/s} \]

### Step 5: Calculate the angle of impact
To find the angle \(\theta\) at which the raindrops strike the windshield, we can use the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|\text{velocity in } \hat{j}|}{|\text{velocity in } \hat{i}|} \]
Substituting the values:
\[ \tan(\theta) = \frac{20}{11.11} \]

### Step 6: Calculate \(\theta\)
Now, we can find \(\theta\) using the arctangent function:
\[ \theta = \tan^{-1}\left(\frac{20}{11.11}\right) \]

### Step 7: Final Calculation
Using a calculator:
\[ \theta \approx \tan^{-1}(1.8) \approx 60.95^\circ \]

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

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