The frequency of the whistle of an engine appears to drop to `(2)/(3)rd` of its actual value when it passes a stationary observer. Velocity of sound in air is 330 m/s then the speed of engine is

Options

A. 330 m/s
B. 165 m/s
C. 220 m/s
D. 110 m/s

Answer

To solve the problem, we will use the Doppler effect formula for sound. Here's a step-by-step solution:

### Step 1: Understand the given information
- The frequency of the whistle of the engine appears to drop to \( \frac{2}{3} \) of its actual value when it passes a stationary observer.
- The velocity of sound in air \( v = 330 \, \text{m/s} \).
- Let the actual frequency of the whistle be \( f \).
- The apparent frequency \( f' = \frac{2}{3} f \).

### Step 2: Write the Doppler effect formula
The formula for the apparent frequency \( f' \) when the source is moving away from a stationary observer is given by:

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

Where:
- \( f' \) = apparent frequency
- \( f \) = actual frequency
- \( v \) = speed of sound in air
- \( v_o \) = speed of the observer (0 m/s since the observer is stationary)
- \( v_s \) = speed of the source (engine)

### Step 3: Substitute the known values into the formula
Since the observer is stationary, \( v_o = 0 \). Thus, the formula simplifies to:

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

Substituting \( f' = \frac{2}{3} f \):

\[ \frac{2}{3} f = f \left( \frac{330}{330 - v_s} \right) \]

### Step 4: Cancel \( f \) from both sides
Assuming \( f \neq 0 \):

\[ \frac{2}{3} = \frac{330}{330 - v_s} \]

### Step 5: Cross-multiply to solve for \( v_s \)
Cross-multiplying gives:

\[ 2(330 - v_s) = 3 \times 330 \]

Expanding this:

\[ 660 - 2v_s = 990 \]

### Step 6: Rearrange the equation to isolate \( v_s \)
Rearranging gives:

\[ -2v_s = 990 - 660 \]

\[ -2v_s = 330 \]

Dividing both sides by -2:

\[ v_s = -\frac{330}{2} = -165 \, \text{m/s} \]

Since speed cannot be negative, we take the magnitude:

\[ v_s = 165 \, \text{m/s} \]

### Conclusion
The speed of the engine is \( 165 \, \text{m/s} \).