An observer moving towards a stationary source observes that the apparent frequency is three times the actual frequency of the source. If the velocity of sound is V, the velocity of the observer would be-

Options

A. 4V
B. V
C. 2V
D. `V//2`

Answer

To solve the problem, we will use the Doppler effect formula for sound waves. The apparent frequency observed by a moving observer towards a stationary source can be expressed as:

\[ F' = F \times \frac{V + V_o}{V + V_s} \]

Where:
- \( F' \) is the apparent frequency,
- \( F \) is the actual frequency of the source,
- \( V \) is the velocity of sound,
- \( V_o \) is the velocity of the observer,
- \( V_s \) is the velocity of the source.

Given:
- The observer is moving towards the source, so \( V_s = 0 \) (the source is stationary).
- The apparent frequency \( F' \) is three times the actual frequency \( F \), so \( F' = 3F \).

### Step-by-step solution:

1. **Set up the Doppler effect equation**:
Since the source is stationary, the equation simplifies to:
\[ F' = F \times \frac{V + V_o}{V} \]

2. **Substitute the given values**:
Substitute \( F' = 3F \) into the equation:
\[ 3F = F \times \frac{V + V_o}{V} \]

3. **Cancel \( F \) from both sides**:
Assuming \( F \neq 0 \), we can divide both sides by \( F \):
\[ 3 = \frac{V + V_o}{V} \]

4. **Multiply both sides by \( V \)**:
\[ 3V = V + V_o \]

5. **Rearrange to solve for \( V_o \)**:
Subtract \( V \) from both sides:
\[ 3V - V = V_o \]
\[ V_o = 2V \]

### Final Answer:
The velocity of the observer \( V_o \) is \( 2V \).