Two sources A and B are sounding notes of frequency 680 Hz. A listener moves from A to B with a constant velocity u. If the speed of sound is 340 m/s, What must be the value of u so that he hears 10 beats per second?

Options

A. `2.0(m)/(s)`
B. `2.5(m)/(s)`
C. `30(m)/(s)`
D. `3.5(m)/(s)`

Explanation

To solve the problem, we need to find the value of the listener's velocity \( u \) such that he hears 10 beats per second when moving from source A to source B, both emitting sound at a frequency of 680 Hz. The speed of sound in air is given as 340 m/s.

### Step-by-Step Solution:

1. **Identify the Frequencies:**
- Both sources A and B emit a frequency \( f_0 = 680 \, \text{Hz} \).

2. **Determine the Apparent Frequencies:**
- When the listener moves towards source B, the apparent frequency \( f_B \) heard from source B is given by the formula:
\[ f_B = f_0 \left( \frac{v + u}{v} \right) \]
where \( v \) is the speed of sound (340 m/s) and \( u \) is the listener's speed.

- When the listener moves away from source A, the apparent frequency \( f_A \) heard from source A is given by:
\[ f_A = f_0 \left( \frac{v - u}{v} \right) \]

3. **Substituting Values:**
- Substitute \( f_0 = 680 \, \text{Hz} \) and \( v = 340 \, \text{m/s} \) into the equations:
\[ f_B = 680 \left( \frac{340 + u}{340} \right) = 680 \left( 1 + \frac{u}{340} \right) \]
\[ f_A = 680 \left( \frac{340 - u}{340} \right) = 680 \left( 1 - \frac{u}{340} \right) \]

4. **Calculate the Beat Frequency:**
- The beat frequency \( \Delta f \) is given by:
\[ \Delta f = f_B - f_A \]
- Substitute the expressions for \( f_B \) and \( f_A \):
\[ \Delta f = 680 \left( 1 + \frac{u}{340} \right) - 680 \left( 1 - \frac{u}{340} \right) \]
- Simplifying this gives:
\[ \Delta f = 680 \left( \frac{u}{340} + \frac{u}{340} \right) = 680 \left( \frac{2u}{340} \right) = 4u \]

5. **Set the Beat Frequency to 10 Hz:**
- We know from the problem that the beat frequency is 10 Hz:
\[ 4u = 10 \]

6. **Solve for \( u \):**
- Rearranging gives:
\[ u = \frac{10}{4} = 2.5 \, \text{m/s} \]

### Final Answer:
The value of \( u \) must be \( 2.5 \, \text{m/s} \).

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