Two different sound sources `s_(1) and s_(2)` have frequencies ratio 1:2. Source s, is approaching towards an observer and `s_(2)` is receding from the same observer. Speeds of both `s_(1) and s_(2)` are the same and equal to v. speed of sound in air in 300 m/s. If no beats are heard by the observer the value of V is 1

Options

A. 125 m/s
B. 100 m/s
C. 75 m/s
D. 50 m/s

Answer

To solve the problem, we will use the Doppler effect to find the speed \( V \) of the sound sources \( s_1 \) and \( s_2 \) given their frequency ratio and the conditions of the observer.

### Step-by-Step Solution:

1. **Identify the Frequencies**:
Let the frequency of source \( s_1 \) be \( f \) and the frequency of source \( s_2 \) be \( 2f \) (since the ratio is 1:2).

2. **Doppler Effect for Approaching Source**:
When source \( s_1 \) is approaching the observer, the observed frequency \( f' \) can be calculated using the Doppler effect formula:
\[ f' = f \frac{v}{v - V} \]
where \( v \) is the speed of sound in air (300 m/s) and \( V \) is the speed of source \( s_1 \).

3. **Doppler Effect for Receding Source**:
For source \( s_2 \) which is receding from the observer, the observed frequency \( f'' \) is given by:
\[ f'' = 2f \frac{v}{v + V} \]

4. **Condition for No Beats**:
The condition for no beats to be heard by the observer is that the observed frequencies must be equal:
\[ f' = f'' \]
Substituting the expressions for \( f' \) and \( f'' \):
\[ f \frac{v}{v - V} = 2f \frac{v}{v + V} \]

5. **Canceling \( f \)**:
Since \( f \) is common on both sides, we can cancel it out (assuming \( f \neq 0 \)):
\[ \frac{v}{v - V} = 2 \frac{v}{v + V} \]

6. **Cross Multiplying**:
Cross multiplying gives:
\[ v(v + V) = 2v(v - V) \]

7. **Expanding Both Sides**:
Expanding both sides results in:
\[ v^2 + vV = 2v^2 - 2vV \]

8. **Rearranging the Equation**:
Rearranging gives:
\[ vV + 2vV = 2v^2 - v^2 \]
Simplifying:
\[ 3vV = v^2 \]

9. **Solving for \( V \)**:
Dividing both sides by \( v \) (assuming \( v \neq 0 \)):
\[ 3V = v \]
Thus,
\[ V = \frac{v}{3} \]

10. **Substituting the Value of \( v \)**:
Given \( v = 300 \, \text{m/s} \):
\[ V = \frac{300}{3} = 100 \, \text{m/s} \]

### Final Answer:
The value of \( V \) is \( 100 \, \text{m/s} \).