A policemen blows a whistle of frequency 330 Hz as a car speeds and passed him with avelocity 18 km per hour. Find the change in frequency as heard by the driver of the car just as he passes the police man (Velocity of sound = 320 m//s)

Answer

To solve the problem of finding the change in frequency as heard by the driver of the car when passing the policeman, we will use the Doppler effect formula. Here are the steps to arrive at the solution:

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as 18 km/h. We need to convert this to meters per second (m/s) using the conversion factor \( \frac{5}{18} \).

\[ \text{Speed of car} = 18 \, \text{km/h} \times \frac{5}{18} = 5 \, \text{m/s} \]

### Step 2: Identify the parameters
- Frequency of the whistle (source frequency, \( f_0 \)) = 330 Hz
- Velocity of sound (\( v \)) = 320 m/s
- Velocity of the car (observer, \( v_o \)) = 5 m/s

### Step 3: Calculate the frequency heard by the driver as the car approaches the policeman
When the car is approaching the source (policeman), the formula for the observed frequency (\( f_1 \)) is:

\[ f_1 = f_0 \left( \frac{v + v_o}{v} \right) \]

Substituting the values:

\[ f_1 = 330 \left( \frac{320 + 5}{320} \right) = 330 \left( \frac{325}{320} \right) \]

Calculating \( f_1 \):

\[ f_1 = 330 \times 1.015625 \approx 335.16 \, \text{Hz} \]

### Step 4: Calculate the frequency heard by the driver as the car moves away from the policeman
When the car is moving away from the source, the formula for the observed frequency (\( f_2 \)) is:

\[ f_2 = f_0 \left( \frac{v - v_o}{v} \right) \]

Substituting the values:

\[ f_2 = 330 \left( \frac{320 - 5}{320} \right) = 330 \left( \frac{315}{320} \right) \]

Calculating \( f_2 \):

\[ f_2 = 330 \times 0.984375 \approx 324.84 \, \text{Hz} \]

### Step 5: Calculate the change in frequency
The change in frequency (\( \Delta f \)) as heard by the driver is given by:

\[ \Delta f = f_2 - f_1 \]

Substituting the values:

\[ \Delta f = 324.84 - 335.16 = -10.32 \, \text{Hz} \]

### Conclusion
The change in frequency as heard by the driver of the car just as he passes the policeman is approximately -10.32 Hz, indicating a decrease in frequency.

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