Two sources A and B are sounding notes of frequency 660 Hz. A listener moves from A to B with a constant velocity u. If the speed is 330 m/s , what must be the value of u so that he hears 8 beats 8 per second ?

Options

A. 2.8 m/s
B. 2 m/
C. 3.0 m/s
D. 3.5 m/s

Answer

To solve the problem, we need to determine the velocity \( u \) of the listener moving from source A to source B, such that he hears 8 beats per second when both sources are emitting sound at a frequency of 660 Hz.

### Step-by-Step Solution:

1. **Understanding Beats**:
The number of beats heard per second is given by the difference in the frequencies of the two sources. In this case, the listener will hear beats due to the Doppler effect as he moves towards source B.

2. **Doppler Effect for Source A**:
When the listener is moving towards source A, the apparent frequency \( f' \) heard from source A can be calculated using the formula:
\[ f' = f \frac{v - u}{v} \]
where:
- \( f = 660 \, \text{Hz} \) (frequency of source A and B)
- \( v = 330 \, \text{m/s} \) (speed of sound)
- \( u \) = velocity of the listener (to be determined)

3. **Doppler Effect for Source B**:
When the listener is moving towards source B, the apparent frequency \( f'' \) heard from source B is:
\[ f'' = f \frac{v + u}{v} \]

4. **Finding the Beat Frequency**:
The beat frequency \( \Delta f \) is given as:
\[ \Delta f = f'' - f' = 8 \, \text{Hz} \]
Substituting the expressions for \( f' \) and \( f'' \):
\[ \Delta f = \left( f \frac{v + u}{v} \right) - \left( f \frac{v - u}{v} \right) \]
Simplifying this:
\[ \Delta f = f \left( \frac{(v + u) - (v - u)}{v} \right) = f \left( \frac{2u}{v} \right) \]

5. **Substituting Known Values**:
We know \( \Delta f = 8 \, \text{Hz} \) and \( f = 660 \, \text{Hz} \):
\[ 8 = 660 \left( \frac{2u}{330} \right) \]

6. **Solving for \( u \)**:
Rearranging the equation:
\[ 8 = \frac{1320u}{330} \]
Simplifying gives:
\[ 8 = 4u \implies u = \frac{8}{4} = 2 \, \text{m/s} \]

### Final Answer:
The value of \( u \) must be **2 m/s**.