A person carrying a whistle emitting continuously a note of `272 Hz` is runnig towards a reflecting surface with a speed of `18km//h`. The speed of sound in air is `345ms^(-1)` The number of beats heard by him is

Options

A. 4
B. 6
C. 8
D. 0

Answer

To solve the problem, we need to find the number of beats heard by a person carrying a whistle emitting a frequency of 272 Hz while running towards a reflecting surface. The speed of sound in air is given as 345 m/s, and the person is running at a speed of 18 km/h.

### Step-by-Step Solution:

1. **Convert the speed of the person from km/h to m/s:**
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]
\[ \text{Speed} = 18 \times \frac{5}{18} = 5 \text{ m/s} \]

2. **Calculate the frequency of the sound heard by the wall (f'):**
Using the Doppler effect formula for the frequency observed by a stationary observer (the wall):
\[ f' = f \left(\frac{v + v_o}{v - v_s}\right) \]
Here, \( f = 272 \text{ Hz} \), \( v = 345 \text{ m/s} \), \( v_o = 0 \) (the wall is stationary), and \( v_s = 5 \text{ m/s} \) (the person is moving towards the wall).
\[ f' = 272 \left(\frac{345 + 0}{345 - 5}\right) = 272 \left(\frac{345}{340}\right) \]

3. **Calculate f':**
\[ f' = 272 \times 1.0147 \approx 276.0 \text{ Hz} \]

4. **Calculate the frequency of the echo heard by the person (f2):**
Now, the wall acts as a source of sound, and the person is the observer moving towards the wall. The frequency heard by the person (f2) is given by:
\[ f_2 = f' \left(\frac{v + v_o}{v - v_s}\right) \]
Here, \( v_o = 5 \text{ m/s} \) (the person is moving towards the wall), and \( v_s = 0 \) (the wall is stationary).
\[ f_2 = f' \left(\frac{345 + 5}{345 - 0}\right) = f' \left(\frac{350}{345}\right) \]
Substituting \( f' \approx 276.0 \text{ Hz} \):
\[ f_2 = 276.0 \left(\frac{350}{345}\right) \approx 276.0 \times 1.0145 \approx 280.0 \text{ Hz} \]

5. **Calculate the beat frequency (f_beat):**
The beat frequency is given by the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_1 - f_2| \]
Where \( f_1 = 272 \text{ Hz} \) (the frequency of the whistle).
\[ f_{\text{beat}} = |272 - 280| = 8 \text{ Hz} \]

### Final Answer:
The number of beats heard by the person is **8 Hz**.