A source of sound S is moving with a velocity of `50m//s` towards a stationary observer. The observer measures the frequency of the source as 1000 Hz. What will be the apparent frequency of the source as 1000 Hz. What will be the apparent frequency of the source when it is moving away from the observer after crossing him? The velocity of the sound in the medium is `350m//s`

Options

A. 750 Hz
B. 857 Hz
C. 1143 Hz
D. 1333 Hz

Explanation

To solve the problem step by step, we will use the Doppler Effect formula for sound waves.

### Step 1: Understand the Doppler Effect Formula
The formula for the apparent frequency \( f' \) when the source is moving towards a stationary observer is given by:
\[ f' = f \left( \frac{v + v_o}{v - v_s} \right) \]
Where:
- \( f' \) = apparent frequency
- \( f \) = actual frequency of the source
- \( v \) = speed of sound in the medium
- \( v_o \) = speed of the observer (0 m/s since the observer is stationary)
- \( v_s \) = speed of the source

### Step 2: Calculate the Original Frequency
Given that the apparent frequency \( f' \) when the source is approaching is 1000 Hz, the speed of sound \( v \) is 350 m/s, and the speed of the source \( v_s \) is 50 m/s, we can rearrange the formula to find the original frequency \( f \):
\[ 1000 = f \left( \frac{350 + 0}{350 - 50} \right) \]
This simplifies to:
\[ 1000 = f \left( \frac{350}{300} \right) \]
Now, solving for \( f \):
\[ f = 1000 \times \frac{300}{350} \]
\[ f = 1000 \times \frac{6}{7} \approx 857.14 \text{ Hz} \]

### Step 3: Calculate the Apparent Frequency When the Source is Moving Away
Now, we need to find the apparent frequency when the source is moving away from the observer. In this case, the formula becomes:
\[ f' = f \left( \frac{v + v_o}{v + v_s} \right) \]
Substituting the known values:
\[ f' = 857.14 \left( \frac{350 + 0}{350 + 50} \right) \]
This simplifies to:
\[ f' = 857.14 \left( \frac{350}{400} \right) \]
Calculating this gives:
\[ f' = 857.14 \times 0.875 \approx 750 \text{ Hz} \]

### Final Answer
The apparent frequency of the source when it is moving away from the observer is approximately **750 Hz**.

---