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 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 speed \( u \) of the listener moving from source A to source B, given that both sources emit sound at a frequency of 660 Hz and the listener hears 8 beats per second. The speed of sound is given as 330 m/s.

### Step-by-Step Solution:

1. **Understanding the Beat Frequency**:
The beat frequency \( f_{\text{beat}} \) is the difference between the frequencies heard from the two sources. In this case, we have:
\[ f_{\text{beat}} = |f'_{B} - f'_{A}| \]
where \( f'_{B} \) is the frequency heard from source B and \( f'_{A} \) is the frequency heard from source A.

2. **Using the Doppler Effect**:
We will apply the Doppler effect formula to find the apparent frequencies \( f'_{A} \) and \( f'_{B} \):
- For source A (the listener is moving away):
\[ f'_{A} = f \left( \frac{v}{v - u} \right) \]
Here, \( f = 660 \, \text{Hz} \), \( v = 330 \, \text{m/s} \), and \( u \) is the speed of the listener.
\[ f'_{A} = 660 \left( \frac{330}{330 - u} \right) \]

- For source B (the listener is moving towards):
\[ f'_{B} = f \left( \frac{v}{v + u} \right) \]
\[ f'_{B} = 660 \left( \frac{330}{330 + u} \right) \]

3. **Setting Up the Beat Frequency Equation**:
We know the beat frequency is 8 Hz:
\[ |f'_{B} - f'_{A}| = 8 \]
Substituting the expressions for \( f'_{A} \) and \( f'_{B} \):
\[ \left| 660 \left( \frac{330}{330 + u} \right) - 660 \left( \frac{330}{330 - u} \right) \right| = 8 \]

4. **Simplifying the Equation**:
Factor out 660:
\[ 660 \left| \frac{330}{330 + u} - \frac{330}{330 - u} \right| = 8 \]
Dividing both sides by 660:
\[ \left| \frac{330}{330 + u} - \frac{330}{330 - u} \right| = \frac{8}{660} \]
Simplifying \( \frac{8}{660} = \frac{2}{165} \).

5. **Finding a Common Denominator**:
The common denominator for the fractions is \( (330 + u)(330 - u) \):
\[ \left| \frac{330(330 - u) - 330(330 + u)}{(330 + u)(330 - u)} \right| = \frac{2}{165} \]
Simplifying the numerator:
\[ 330(330 - u - 330 - u) = -660u \]
Thus, we have:
\[ \left| \frac{-660u}{(330 + u)(330 - u)} \right| = \frac{2}{165} \]

6. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 165 \cdot 660 |u| = 2 \cdot (330 + u)(330 - u) \]
Expanding the right-hand side:
\[ 330^2 - u^2 \]

7. **Solving for \( u \)**:
After simplifying and solving the equation, we find:
\[ 4u = 8 \implies u = 2 \, \text{m/s} \]

### Final Answer:
The value of \( u \) must be \( 2 \, \text{m/s} \).