The frequency of a source of sound as measured by an observer when the source is moving towards him with a speed of 30 m/s is 720 Hz. The apparent frequency when the source is moving away after crossing the observer is `x xx 10^2 Hz` what is the value of x. ......... (velocity of sound is 330 m/s)

Answer

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

### Step 1: Identify the known values
- Speed of sound, \( v = 330 \, \text{m/s} \)
- Speed of the source, \( v_s = 30 \, \text{m/s} \)
- Apparent frequency when the source is moving towards the observer, \( f_1 = 720 \, \text{Hz} \)

### Step 2: Use the Doppler effect formula for the source moving towards the observer
The formula for the apparent frequency when the source is moving towards the observer is given by:
\[ f_1 = \frac{v}{v - v_s} f_0 \]
Where \( f_0 \) is the actual frequency of the source.

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 720 = \frac{330}{330 - 30} f_0 \]
This simplifies to:
\[ 720 = \frac{330}{300} f_0 \]

### Step 4: Solve for the actual frequency \( f_0 \)
Rearranging the equation to find \( f_0 \):
\[ f_0 = 720 \times \frac{300}{330} \]
Calculating \( f_0 \):
\[ f_0 = 720 \times \frac{300}{330} = 720 \times \frac{10}{11} = 7200 / 11 \approx 654.55 \, \text{Hz} \]

### Step 5: Use the Doppler effect formula for the source moving away from the observer
Now, when the source is moving away from the observer, the formula for the apparent frequency is:
\[ f_2 = \frac{v}{v + v_s} f_0 \]
We need to find \( f_2 \) which is given as \( x \times 10^2 \, \text{Hz} \).

### Step 6: Substitute \( f_0 \) into the formula for \( f_2 \)
Substituting the values into the formula:
\[ f_2 = \frac{330}{330 + 30} f_0 = \frac{330}{360} f_0 \]
Substituting \( f_0 \):
\[ f_2 = \frac{330}{360} \times \frac{7200}{11} \]

### Step 7: Calculate \( f_2 \)
Calculating \( f_2 \):
\[ f_2 = \frac{330 \times 7200}{360 \times 11} = \frac{2376000}{3960} \approx 600 \, \text{Hz} \]

### Step 8: Express \( f_2 \) in terms of \( x \)
Since \( f_2 = x \times 10^2 \):
\[ 600 = x \times 10^2 \]
Thus, solving for \( x \):
\[ x = \frac{600}{100} = 6 \]

### Final Answer
The value of \( x \) is \( 6 \).