A and B are two identical sound sources cachemitting a note of frequency 1000 Hz. A person moves from A to B along the line joining the two sources. How fast should be move to that he has 7 beats per second. velocity of sound 330 m/s ?

Answer

To solve the problem step by step, we will use the concept of the Doppler effect and the information provided in the question.

### Step 1: Understand the Problem
We have two identical sound sources (A and B) emitting sound at a frequency of 1000 Hz. A person is moving from source A to source B and we need to find out how fast he should move to hear 7 beats per second.

### Step 2: Define the Variables
- Frequency of the sound sources, \( f_0 = 1000 \, \text{Hz} \)
- Velocity of sound, \( v = 330 \, \text{m/s} \)
- Beats per second, \( \Delta f = 7 \, \text{Hz} \)

### Step 3: Apply the Doppler Effect
The frequency heard by the observer moving towards source A and away from source B can be calculated using the Doppler effect formula.

For source A (moving towards):
\[ f_A = \frac{v + v_o}{v} f_0 \]
For source B (moving away):
\[ f_B = \frac{v - v_o}{v} f_0 \]

### Step 4: Set Up the Equation for Beats
The difference in frequencies (beats) is given by:
\[ |f_A - f_B| = 7 \]
This implies:
\[ f_A - f_B = 7 \]

### Step 5: Substitute the Frequencies
Substituting \( f_A \) and \( f_B \) into the beats equation:
\[ \left(\frac{330 + v_o}{330} \cdot 1000\right) - \left(\frac{330 - v_o}{330} \cdot 1000\right) = 7 \]

### Step 6: Simplify the Equation
This simplifies to:
\[ \frac{(330 + v_o) \cdot 1000 - (330 - v_o) \cdot 1000}{330} = 7 \]
\[ \frac{1000(330 + v_o - 330 + v_o)}{330} = 7 \]
\[ \frac{2000 v_o}{330} = 7 \]

### Step 7: Solve for \( v_o \)
Now, multiply both sides by 330:
\[ 2000 v_o = 7 \cdot 330 \]
\[ 2000 v_o = 2310 \]
\[ v_o = \frac{2310}{2000} = 1.155 \, \text{m/s} \]

### Step 8: Final Answer
The speed at which the person should move is approximately:
\[ v_o \approx 1.15 \, \text{m/s} \]