When a source of frequency `f_(0)` moves away from a stationary observer with a certain velocity, an apparent frequency `f'` is observed. When it moves with the same speed towards the observer, the observed frequecy is `1.2 f'`. If the velocity of sound is v, then the acutual frequency `f_(0)` is

Options

A. `12/11 f'`
B. `11/12 f'`
C. `7/6 f'`
D. `6/7 f'`

Answer

To solve the problem, we will use the Doppler effect formula for sound. The apparent frequency observed when a source is moving away from a stationary observer and when it is moving towards the observer can be expressed using the following equations.

### Step-by-Step Solution:

1. **Understanding the Doppler Effect**:
- When a source of sound moves away from a stationary observer, the apparent frequency \( f' \) is given by:
\[ f' = \frac{v}{v + v_s} f_0 \]
- When the source moves towards the observer, the apparent frequency is given by:
\[ f'' = \frac{v}{v - v_s} f_0 \]

2. **Setting Up the Equations**:
- From the problem statement, we know:
\[ f'' = 1.2 f' \]
- Substituting the expressions for \( f' \) and \( f'' \):
\[ \frac{v}{v - v_s} f_0 = 1.2 \left(\frac{v}{v + v_s} f_0\right) \]

3. **Canceling \( f_0 \)**:
- Since \( f_0 \) is common in both sides and is not zero, we can cancel it out:
\[ \frac{v}{v - v_s} = 1.2 \left(\frac{v}{v + v_s}\right) \]

4. **Cross Multiplying**:
- Cross-multiplying gives us:
\[ v(v + v_s) = 1.2 v (v - v_s) \]

5. **Expanding Both Sides**:
- Expanding both sides results in:
\[ v^2 + v v_s = 1.2 v^2 - 1.2 v v_s \]

6. **Rearranging the Equation**:
- Bringing all terms involving \( v_s \) to one side and constant terms to the other side:
\[ v v_s + 1.2 v v_s = 1.2 v^2 - v^2 \]
- This simplifies to:
\[ 2.2 v_s = 0.2 v^2 \]

7. **Solving for \( v_s \)**:
- Dividing both sides by 2.2:
\[ v_s = \frac{0.2 v^2}{2.2} = \frac{v^2}{11} \]

8. **Substituting \( v_s \) Back into the Equation for \( f' \)**:
- Now, we substitute \( v_s \) back into the equation for \( f' \):
\[ f' = \frac{v}{v + v_s} f_0 = \frac{v}{v + \frac{v^2}{11}} f_0 \]
- This can be simplified to:
\[ f' = \frac{v}{\frac{11v + v^2}{11}} f_0 = \frac{11v}{11v + v^2} f_0 \]

9. **Finding \( f_0 \)**:
- Rearranging gives us:
\[ f_0 = \frac{11v}{11v + v^2} f' \]
- This can be simplified to:
\[ f_0 = \frac{12}{11} f' \]

### Final Answer:
Thus, the actual frequency \( f_0 \) is:
\[ f_0 = \frac{12}{11} f' \]