A whistle emitting a sound of frequency 450 Hz approaches a stationary observer at a speed of 33 m/s. Velocity of sound is 330 m/s. The frequency heard by the observer, in Hz, is :

Options

A. 409
B. 429
C. 517
D. 500

Explanation

To solve the problem of finding the frequency heard by a stationary observer when a whistle emitting a sound of frequency 450 Hz approaches at a speed of 33 m/s, we can use the Doppler effect formula. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Frequency of the whistle (emitted frequency, \( f \)) = 450 Hz
- Speed of the whistle (source) (\( v_s \)) = 33 m/s (approaching)
- Speed of sound in air (\( v \)) = 330 m/s
- Speed of the observer (\( v_o \)) = 0 m/s (since the observer is stationary)

### Step 2: Write the Doppler effect formula
The formula for the frequency heard by the observer when the source is moving towards a stationary observer is given by:

\[ f' = \frac{v + v_o}{v - v_s} \cdot f \]

Where:
- \( f' \) = frequency heard by the observer
- \( v \) = speed of sound
- \( v_o \) = speed of the observer
- \( v_s \) = speed of the source
- \( f \) = emitted frequency

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

\[ f' = \frac{330 + 0}{330 - 33} \cdot 450 \]

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

\[ f' = \frac{330}{297} \cdot 450 \]

### Step 5: Calculate the fraction
Now, simplify \( \frac{330}{297} \):

\[ \frac{330}{297} = \frac{10}{9} \]

### Step 6: Substitute back into the equation
Now substitute this back into the equation for \( f' \):

\[ f' = \frac{10}{9} \cdot 450 \]

### Step 7: Calculate \( f' \)
Now calculate \( f' \):

\[ f' = \frac{4500}{9} = 500 \text{ Hz} \]

### Conclusion
The frequency heard by the observer is **500 Hz**.

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