A stationary source emits sound waves of frequency 500 Hz. Two observes moving along a line passing through the source detect sount to be of frequencies 480 Hz and 530 Hz.Their respective speeds are, in `ms^(-1)`
(Given speed of sound `=300 m//s`)

Options

A. 12, 16
B. 12, 18
C. 16, 14
D. 8, 18

Answer

To solve the problem, we will use the Doppler effect formula for sound waves. The frequency observed by a moving observer is given by:

\[ f' = f_0 \left( \frac{v + v_o}{v} \right) \]

where:
- \( f' \) is the observed frequency,
- \( f_0 \) is the source frequency,
- \( v \) is the speed of sound,
- \( v_o \) is the speed of the observer (positive if moving towards the source, negative if moving away).

### Step 1: Setting Up the Equations

1. For Observer A (detects frequency \( f_A = 480 \, \text{Hz} \)):
Since the frequency is lower, Observer A is moving away from the source. Thus, we use the negative sign for \( v_o \):
\[ 480 = 500 \left( \frac{300 - v_A}{300} \right) \]

2. For Observer B (detects frequency \( f_B = 530 \, \text{Hz} \)):
Since the frequency is higher, Observer B is moving towards the source. Thus, we use the positive sign for \( v_o \):
\[ 530 = 500 \left( \frac{300 + v_B}{300} \right) \]

### Step 2: Solving for \( v_A \)

From the equation for Observer A:
\[ 480 = 500 \left( \frac{300 - v_A}{300} \right) \]
Multiply both sides by 300:
\[ 480 \times 300 = 500 (300 - v_A) \]
\[ 144000 = 150000 - 500 v_A \]
Rearranging gives:
\[ 500 v_A = 150000 - 144000 \]
\[ 500 v_A = 6000 \]
\[ v_A = \frac{6000}{500} = 12 \, \text{m/s} \]

### Step 3: Solving for \( v_B \)

From the equation for Observer B:
\[ 530 = 500 \left( \frac{300 + v_B}{300} \right) \]
Multiply both sides by 300:
\[ 530 \times 300 = 500 (300 + v_B) \]
\[ 159000 = 150000 + 500 v_B \]
Rearranging gives:
\[ 500 v_B = 159000 - 150000 \]
\[ 500 v_B = 9000 \]
\[ v_B = \frac{9000}{500} = 18 \, \text{m/s} \]

### Final Results

The speeds of the observers are:
- \( v_A = 12 \, \text{m/s} \) (moving away from the source)
- \( v_B = 18 \, \text{m/s} \) (moving towards the source)

### Summary

The respective speeds of the observers are:
- Observer A: \( 12 \, \text{m/s} \)
- Observer B: \( 18 \, \text{m/s} \)