A siren emitting a sound of frequency 800 Hz moves away from an observer towards a cliff at a speed of `15ms^-1`. Then the frequency of sound that the observer hears in the echo reflected from the cliff is (Take velocity of sound in air `=330ms^-1`)

Options

A. `885 Hz`
B. `765 Hz`
C. `800 Hz`
D. `838 Hz`

Explanation

To solve the problem of finding the frequency of sound that the observer hears in the echo reflected from the cliff, we can break it down into a series of steps.

### Step 1: Identify the Given Information
- Frequency of the siren (f₀) = 800 Hz
- Speed of the siren (vₛ) = 15 m/s (moving away from the observer)
- Speed of sound in air (v) = 330 m/s
- Observer is stationary (vₒ = 0 m/s)

### Step 2: Calculate the Frequency Heard by the Cliff
Since the siren is moving away from the observer, we need to use the Doppler effect formula to find the frequency of sound that the cliff (acting as an observer) hears.

The formula for the observed frequency (f) when the source is moving away from a stationary observer is given by:
\[ f = \frac{v}{v + vₛ} \cdot f₀ \]

Substituting the values:
\[ f = \frac{330}{330 + 15} \cdot 800 \]
\[ f = \frac{330}{345} \cdot 800 \]
\[ f = 0.9565 \cdot 800 \]
\[ f \approx 765.2 \text{ Hz} \]

### Step 3: Calculate the Frequency of the Echo Heard by the Observer
Now, the cliff reflects the sound back, and the observer will hear this reflected sound. In this case, the cliff acts as a source of sound, and the observer is stationary.

Using the Doppler effect formula again for the frequency heard by the observer from the cliff:
\[ f' = \frac{v + vₒ}{v - vₛ} \cdot f \]

Since the observer is stationary (vₒ = 0), the formula simplifies to:
\[ f' = \frac{v}{v - vₛ} \cdot f \]

Substituting the values:
\[ f' = \frac{330}{330 - 15} \cdot 765.2 \]
\[ f' = \frac{330}{315} \cdot 765.2 \]
\[ f' = 1.0476 \cdot 765.2 \]
\[ f' \approx 800 \text{ Hz} \]

### Final Result
The frequency of sound that the observer hears in the echo reflected from the cliff is approximately **800 Hz**.

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