An engine moving away from a vertical cliff blows a born at a frequency f. Its speed is 0.5% of the speed of sound in air. The frequency of the reflected sound received at the engine is

Options

A. 0.990 f
B. 0.995 f
C. 1.005
D. 1.010 f

Explanation

To solve the problem of finding the frequency of the reflected sound received at the engine, we can use the principles of the Doppler effect. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the scenario
The engine is moving away from a vertical cliff and emits a sound at frequency \( f \). The speed of the engine is given as \( 0.5\% \) of the speed of sound in air, denoted as \( c \).

### Step 2: Calculate the speed of the engine
The speed of the engine \( v_e \) can be calculated as:
\[ v_e = 0.005c \]

### Step 3: Determine the frequency of sound reaching the cliff
When the sound travels from the engine to the cliff, the cliff acts as a stationary observer, and the engine is the source moving away. The formula for the apparent frequency \( f' \) observed at the cliff is given by:
\[ f' = f \left( \frac{c}{c + v_e} \right) \]
Substituting the values, we have:
\[ f' = f \left( \frac{c}{c + 0.005c} \right) = f \left( \frac{c}{1.005c} \right) = \frac{f}{1.005} \]

### Step 4: Determine the frequency of the reflected sound received at the engine
Now, the sound reflected from the cliff travels back to the engine. In this case, the engine is moving towards the cliff (the source of the reflected sound). The formula for the apparent frequency \( f'' \) received at the engine is:
\[ f'' = f' \left( \frac{c + v_e}{c} \right) \]
Substituting \( f' \) from the previous step:
\[ f'' = \frac{f}{1.005} \left( \frac{c + 0.005c}{c} \right) = \frac{f}{1.005} \left( \frac{1.005c}{c} \right) = \frac{f}{1.005} \cdot 1.005 = f \cdot \frac{1}{1.005} \]

### Step 5: Final calculation
Now we simplify the expression:
\[ f'' = f \cdot \frac{1}{1.005} \approx 0.995 f \]

### Conclusion
The frequency of the reflected sound received at the engine is:
\[ f'' \approx 0.995f \]