A sound source is moving with speed `50 m//s` towards a fixed observer. Frequency observed by observer is `1000Hz`. Find out apparent frequency observed by observer when source is moving away from observer (Speed of sound `=350 m//s`)

Options

A. 750 Hz
B. 857 Hz
C. 1143 Hz
D. 807 Hz

Answer

To solve the problem, we will use the Doppler effect formula for sound. The frequency observed by a stationary observer when the source is moving towards or away from them can be calculated using the following formulas:

1. When the source is moving towards the observer:
\[ f' = f \frac{v}{v - v_s} \]
where:
- \( f' \) = observed frequency
- \( f \) = actual frequency of the source
- \( v \) = speed of sound
- \( v_s \) = speed of the source

2. When the source is moving away from the observer:
\[ f'' = f \frac{v}{v + v_s} \]
where \( f'' \) is the observed frequency when the source is moving away.

### Step-by-Step Solution

**Step 1: Identify the known values.**
- Speed of sound, \( v = 350 \, \text{m/s} \)
- Speed of the source, \( v_s = 50 \, \text{m/s} \)
- Observed frequency when the source is approaching, \( f' = 1000 \, \text{Hz} \)

**Step 2: Use the first formula to find the actual frequency \( f \).**
Using the formula for the frequency when the source is moving towards the observer:
\[ f' = f \frac{v}{v - v_s} \]
Substituting the known values:
\[ 1000 = f \frac{350}{350 - 50} \]
\[ 1000 = f \frac{350}{300} \]
\[ 1000 = f \cdot \frac{7}{6} \]
Now, solve for \( f \):
\[ f = 1000 \cdot \frac{6}{7} = \frac{6000}{7} \approx 857.14 \, \text{Hz} \]

**Step 3: Use the second formula to find the apparent frequency \( f'' \) when the source is moving away.**
Using the formula for the frequency when the source is moving away from the observer:
\[ f'' = f \frac{v}{v + v_s} \]
Substituting the values we have:
\[ f'' = \left(\frac{6000}{7}\right) \frac{350}{350 + 50} \]
\[ f'' = \left(\frac{6000}{7}\right) \frac{350}{400} \]
\[ f'' = \left(\frac{6000}{7}\right) \cdot \frac{7}{8} = \frac{6000}{8} = 750 \, \text{Hz} \]

### Final Answer:
The apparent frequency observed by the observer when the source is moving away from them is \( 750 \, \text{Hz} \).