At what speed should a source of sound move so that observer finds the apparent frequency equal to half of the original frequency ?

Options

A. `(v)/(2)`
B. `2 v `
C. `v//4`
D. v

Explanation

To solve the problem of finding the speed at which a source of sound must move so that the observer finds the apparent frequency equal to half of the original frequency, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information**:
- Let the original frequency be \( f \).
- The apparent frequency \( f' \) is given as \( \frac{f}{2} \).
- The speed of sound in air is denoted as \( v \).
- The velocity of the observer \( v_0 = 0 \) (the observer is stationary).

2. **Use the Doppler Effect Formula**:
The formula for the apparent frequency when the source is moving and the observer is stationary is given by:
\[ f' = \frac{f (v - v_0)}{v - v_s} \]
Here, \( v_s \) is the speed of the source.

3. **Substitute Known Values**:
Since \( v_0 = 0 \), the formula simplifies to:
\[ f' = \frac{f v}{v - v_s} \]
We know \( f' = \frac{f}{2} \), so we can substitute this into the equation:
\[ \frac{f}{2} = \frac{f v}{v - v_s} \]

4. **Eliminate \( f \)**:
We can cancel \( f \) from both sides (assuming \( f \neq 0 \)):
\[ \frac{1}{2} = \frac{v}{v - v_s} \]

5. **Cross-Multiply**:
Cross-multiplying gives us:
\[ v - v_s = 2v \]

6. **Rearrange the Equation**:
Rearranging the equation leads to:
\[ -v_s = 2v - v \]
\[ -v_s = v \]

7. **Solve for \( v_s \)**:
Thus, we find:
\[ v_s = -v \]
The negative sign indicates that the source is moving away from the observer.

### Final Answer:
The speed at which the source of sound should move is \( v \) meters per second away from the observer.

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