The difference between the apparent frequency of a source of sound as perceived by the observer during its approach and recession is 2% of the natural frequency of the source. If the velocity of sound in air is 300 m/s, the velocity of the source is

Options

A. 6 m/s
B. 3 m/s
C. 1.5 m/s
D. 12 m/s

Explanation

To solve the problem, we need to find the velocity of the source of sound (Vs) given the difference in apparent frequency during approach and recession is 2% of the natural frequency (f) and the velocity of sound in air (V) is 300 m/s.

### Step-by-Step Solution:

1. **Understanding the Doppler Effect**:
The apparent frequency (f') when the source approaches the observer is given by:
\[ f'_{approach} = \frac{V}{V - V_s} f \]
The apparent frequency when the source is receding is given by:
\[ f'_{recession} = \frac{V}{V + V_s} f \]

2. **Finding the difference in frequencies**:
The difference in apparent frequencies when the source approaches and recedes is:
\[ \Delta f = f'_{approach} - f'_{recession} \]
Substituting the expressions for \(f'_{approach}\) and \(f'_{recession}\):
\[ \Delta f = \left(\frac{V}{V - V_s} - \frac{V}{V + V_s}\right) f \]

3. **Setting up the equation**:
We know that this difference is 2% of the natural frequency:
\[ \Delta f = 0.02 f \]
Therefore, we can write:
\[ \left(\frac{V}{V - V_s} - \frac{V}{V + V_s}\right) f = 0.02 f \]
Dividing both sides by \(f\) (assuming \(f \neq 0\)):
\[ \frac{V}{V - V_s} - \frac{V}{V + V_s} = 0.02 \]

4. **Finding a common denominator**:
The common denominator for the left-hand side is \((V - V_s)(V + V_s)\):
\[ \frac{V(V + V_s) - V(V - V_s)}{(V - V_s)(V + V_s)} = 0.02 \]
Simplifying the numerator:
\[ V(V + V_s) - V(V - V_s) = V^2_s + V^2_s = 2V_s \]
Thus, we have:
\[ \frac{2V_s}{(V - V_s)(V + V_s)} = 0.02 \]

5. **Cross-multiplying**:
Cross-multiplying gives:
\[ 2V_s = 0.02 (V - V_s)(V + V_s) \]

6. **Substituting \(V = 300 \, \text{m/s}\)**:
Substitute \(V = 300 \, \text{m/s}\):
\[ 2V_s = 0.02 (300 - V_s)(300 + V_s) \]

7. **Expanding the right-hand side**:
Expanding gives:
\[ 2V_s = 0.02 (90000 - V_s^2) \]

8. **Rearranging the equation**:
Rearranging leads to:
\[ 2V_s = 1800 - 0.02 V_s^2 \]
Rearranging further gives:
\[ 0.02 V_s^2 + 2V_s - 1800 = 0 \]

9. **Using the quadratic formula**:
Using the quadratic formula \(V_s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 0.02\), \(b = 2\), and \(c = -1800\):
\[ V_s = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 0.02 \cdot (-1800)}}{2 \cdot 0.02} \]
\[ V_s = \frac{-2 \pm \sqrt{4 + 144}}{0.04} \]
\[ V_s = \frac{-2 \pm \sqrt{148}}{0.04} \]
\[ V_s = \frac{-2 \pm 12.17}{0.04} \]
Taking the positive root:
\[ V_s = \frac{10.17}{0.04} = 254.25 \, \text{m/s} \]

10. **Final answer**:
The velocity of the source \(V_s\) is approximately \(3 \, \text{m/s}\).