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. `950 Hz`
C. `550 Hz`
D. `350 Hz`

Answer

To solve the problem, we will use the Doppler effect formula for sound. The formula for apparent frequency when the source is moving towards the observer is given by:

\[ f' = f_0 \frac{v + v_o}{v - v_s} \]

Where:
- \( f' \) = apparent frequency
- \( f_0 \) = original frequency (frequency of the source)
- \( v \) = speed of sound in air
- \( v_o \) = speed of the observer (0 if the observer is stationary)
- \( v_s \) = speed of the source

### Step 1: Determine the original frequency (\( f_0 \))

From the problem, we know that when the source is moving towards the observer with a speed of \( 50 \, m/s \), the observed frequency is \( 1000 \, Hz \).

Using the formula for the case when the source is moving towards the observer:

\[ 1000 = f_0 \frac{350 + 0}{350 - 50} \]

### Step 2: Solve for \( f_0 \)

Substituting the values into the equation:

\[ 1000 = f_0 \frac{350}{300} \]

Now, simplify the fraction:

\[ 1000 = f_0 \cdot \frac{7}{6} \]

Now, solve for \( f_0 \):

\[ f_0 = 1000 \cdot \frac{6}{7} = \frac{6000}{7} \approx 857.14 \, Hz \]

### Step 3: Calculate the apparent frequency when the source is moving away from the observer

Now, we need to find the apparent frequency when the source is moving away from the observer. The formula for this case is:

\[ f' = f_0 \frac{v + v_o}{v + v_s} \]

Here, \( v_o = 0 \) (observer is stationary) and \( v_s = 50 \, m/s \).

Substituting the values:

\[ f' = f_0 \frac{350 + 0}{350 + 50} \]

\[ f' = f_0 \frac{350}{400} \]

### Step 4: Substitute \( f_0 \) into the equation

Now substitute \( f_0 \):

\[ f' = \frac{6000}{7} \cdot \frac{350}{400} \]

### Step 5: Simplify the expression

Now simplify:

\[ f' = \frac{6000 \cdot 350}{7 \cdot 400} \]

Calculating the values:

\[ f' = \frac{2100000}{2800} = 750 \, Hz \]

### Final Answer

The apparent frequency observed by the observer when the source is moving away is \( 750 \, Hz \).

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