A tuning fork of frequency `328Hz` is moved towards a wall at a speed of `2ms^(-1)`. An observer standing on the same side as the fork hears two sounds, one directly from the fork and the other reflected from the wall. Number of beats per second is (Velocity of sound in air `330ms^(-1)`).

Options

A. `4`
B. `5`
C. `6`
D. `7`

Explanation

To solve the problem, we need to determine the frequencies of the sounds heard by the observer: one directly from the tuning fork and the other reflected from the wall. We will then find the number of beats per second between these two frequencies.

### Step 1: Determine the frequency heard directly from the tuning fork
The frequency of the tuning fork is given as \( f_0 = 328 \, \text{Hz} \). Since the observer is stationary and the source (tuning fork) is moving towards the observer, we can use the Doppler effect formula for a source moving towards a stationary observer:

\[ f' = f_0 \frac{v + v_o}{v - v_s} \]

Where:
- \( f' \) = frequency heard by the observer
- \( f_0 \) = frequency of the source (328 Hz)
- \( v \) = speed of sound in air (330 m/s)
- \( v_o \) = speed of the observer (0 m/s, since the observer is stationary)
- \( v_s \) = speed of the source (2 m/s)

Substituting the values:

\[ f' = 328 \, \text{Hz} \cdot \frac{330 + 0}{330 - 2} = 328 \cdot \frac{330}{328} = 330 \, \text{Hz} \]

### Step 2: Determine the frequency of the sound reflected from the wall
When the sound reflects off the wall, the wall acts as a stationary source. The frequency heard by the observer from the reflected sound can be calculated using the Doppler effect formula again, but this time the source is the wall (which is stationary) and the observer is moving towards it. The frequency of the sound reflected back to the observer is given by:

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

Where:
- \( f'' \) = frequency of the reflected sound
- \( f' \) = frequency heard from the tuning fork (330 Hz)
- \( v_o \) = speed of the observer (0 m/s)
- \( v_s \) = speed of the source (2 m/s)

Substituting the values:

\[ f'' = 330 \, \text{Hz} \cdot \frac{330 + 0}{330 - 2} = 330 \cdot \frac{330}{328} = 332 \, \text{Hz} \]

### Step 3: Calculate the number of beats per second
The number of beats per second is given by the absolute difference between the two frequencies:

\[ \text{Number of beats} = |f'' - f'| = |332 \, \text{Hz} - 330 \, \text{Hz}| = 2 \, \text{Hz} \]

### Final Answer
The number of beats per second is \( 2 \, \text{Hz} \).