Two engines pass each other moving in opposite directions with uniform speed of 30m/so .one of them is blowing a whistle of frequency 540 Hz. Calculate the frequency heard by driver of second engine before they pass each other. Speed of sound is 300 m/sec :

Options

A. 648 Hz
B. 450 Hz
C. 270 Hz
D. 540 Hz

Answer

To solve the problem of finding the frequency heard by the driver of the second engine before they pass each other, we will use the Doppler effect formula for sound. Here’s a step-by-step solution:

### Step 1: Identify the known values
- Frequency of the whistle (f₀) = 540 Hz
- Speed of sound (v) = 300 m/s
- Speed of the observer (v₀) = 30 m/s (the driver of the second engine)
- Speed of the source (vₛ) = 30 m/s (the engine blowing the whistle)

### Step 2: Apply the Doppler effect formula
The formula for the frequency heard by the observer when the source and observer are moving towards each other is given by:

\[ f' = f_0 \left( \frac{v + v_0}{v - v_s} \right) \]

Where:
- \( f' \) = frequency heard by the observer
- \( f_0 \) = frequency of the source (540 Hz)
- \( v \) = speed of sound (300 m/s)
- \( v_0 \) = speed of the observer (30 m/s)
- \( v_s \) = speed of the source (30 m/s)

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:

\[ f' = 540 \left( \frac{300 + 30}{300 - 30} \right) \]

### Step 4: Simplify the expression
Calculate the numerator and denominator:

- Numerator: \( 300 + 30 = 330 \)
- Denominator: \( 300 - 30 = 270 \)

Now substituting these values back into the equation:

\[ f' = 540 \left( \frac{330}{270} \right) \]

### Step 5: Simplify the fraction
Now simplify \( \frac{330}{270} \):

\[ \frac{330}{270} = \frac{11}{9} \]

### Step 6: Calculate the final frequency
Now substituting back into the equation:

\[ f' = 540 \times \frac{11}{9} \]

Calculating this gives:

\[ f' = 540 \times 1.2222 \approx 660 \text{ Hz} \]

### Conclusion
The frequency heard by the driver of the second engine before they pass each other is **660 Hz**.

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