A source and an observer is approaching closer with a relative velocity of `40 ms^(-1)`. If the true frequency of the source is 1200 Hz calculate the observed frequency under these conditions: (i) the source alone is moving (ii) the observer alone is moving. Take velocity of sound in air to be `340 ms^(-1)`.

Answer

To solve the problem, we will use the Doppler effect formula for sound. The formula for the observed frequency (\( f' \)) when either the source or the observer is moving is given by:

\[ f' = f \frac{v + v_0}{v - v_s} \]

Where:
- \( f' \) is the observed frequency,
- \( f \) is the true frequency of the source,
- \( v \) is the speed of sound in air,
- \( v_0 \) is the speed of the observer (positive if moving towards the source),
- \( v_s \) is the speed of the source (positive if moving away from the observer).

### Part (i): When the source alone is moving

1. **Identify the known values**:
- True frequency of the source, \( f = 1200 \, \text{Hz} \)
- Speed of sound, \( v = 340 \, \text{m/s} \)
- Speed of the source, \( v_s = 40 \, \text{m/s} \) (moving towards the observer)
- Speed of the observer, \( v_0 = 0 \, \text{m/s} \) (observer is stationary)

2. **Substitute the values into the Doppler effect formula**:
\[ f' = 1200 \frac{340 + 0}{340 - 40} \]

3. **Calculate the denominator**:
\[ 340 - 40 = 300 \]

4. **Calculate the observed frequency**:
\[ f' = 1200 \frac{340}{300} = 1200 \times 1.1333 \approx 1360 \, \text{Hz} \]

### Part (ii): When the observer alone is moving

1. **Identify the known values**:
- True frequency of the source, \( f = 1200 \, \text{Hz} \)
- Speed of sound, \( v = 340 \, \text{m/s} \)
- Speed of the observer, \( v_0 = 40 \, \text{m/s} \) (moving towards the source)
- Speed of the source, \( v_s = 0 \, \text{m/s} \) (source is stationary)

2. **Substitute the values into the Doppler effect formula**:
\[ f' = 1200 \frac{340 + 40}{340 - 0} \]

3. **Calculate the numerator**:
\[ 340 + 40 = 380 \]

4. **Calculate the observed frequency**:
\[ f' = 1200 \frac{380}{340} = 1200 \times 1.1176 \approx 1341.2 \, \text{Hz} \]

### Final Answers:
- (i) When the source alone is moving: \( f' \approx 1360 \, \text{Hz} \)
- (ii) When the observer alone is moving: \( f' \approx 1341.2 \, \text{Hz} \)