A police car with a siren of frequency `8 KHz` is moving with uniform velocity `36 Km//hr` towards a ball building which reflects the sound waves. The speed of sound in air is `320 m//s`. The frequency of the siren heard by the car driver is

Options

A. `8.50kHz`
B. `8.25kHz`
C. `7.75kHz`
D. `7.50kHz`

Explanation

To solve the problem, we will use the Doppler effect formula to find the frequency of the siren heard by the car driver. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Values
- Frequency of the siren (source frequency), \( f_0 = 8 \, \text{kHz} = 8000 \, \text{Hz} \)
- Speed of the police car (source), \( v_s = 36 \, \text{km/hr} \)
- Speed of sound in air, \( c = 320 \, \text{m/s} \)

### Step 2: Convert the Speed of the Police Car
Convert the speed from km/hr to m/s:
\[ v_s = 36 \, \text{km/hr} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{hr}}{3600 \, \text{s}} = 10 \, \text{m/s} \]

### Step 3: Calculate the Apparent Frequency at the Building
When the police car approaches the building, the frequency heard by the building (which is stationary) can be calculated using the formula:
\[ f' = f_0 \frac{c}{c - v_s} \]
Substituting the values:
\[ f' = 8000 \, \text{Hz} \cdot \frac{320 \, \text{m/s}}{320 \, \text{m/s} - 10 \, \text{m/s}} = 8000 \cdot \frac{320}{310} \]

### Step 4: Calculate the Frequency
Now, calculate \( f' \):
\[ f' = 8000 \cdot \frac{320}{310} \approx 8000 \cdot 1.03226 \approx 8260.65 \, \text{Hz} \]

### Step 5: Calculate the Frequency Heard by the Driver
Now, the reflected sound wave will be heard by the driver. The driver is now the observer moving towards the building. The formula for the frequency heard by the driver is:
\[ f'' = f' \frac{c + v_o}{c} \]
Where \( v_o \) is the speed of the observer (the driver), which is also \( 10 \, \text{m/s} \):
\[ f'' = 8260.65 \cdot \frac{320 + 10}{320} = 8260.65 \cdot \frac{330}{320} \]

### Step 6: Calculate the Final Frequency
Now, calculate \( f'' \):
\[ f'' = 8260.65 \cdot 1.03125 \approx 8530.65 \, \text{Hz} \approx 8.53 \, \text{kHz} \]

### Conclusion
The frequency of the siren heard by the car driver is approximately **8.53 kHz**.

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