A stationary source is emitting sound at a fixed frequency `f_(0)`, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is `1.2%` of `f_(0)`. What is the difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant speeds much smaller than the speed of sound which is `330 ms^(-1)`.

Options

A. 2
B. 3
C. 5
D. 7

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Doppler Effect
When a stationary source emits sound, and an observer (in this case, the cars) is moving towards the source, the frequency of the sound perceived by the observer is higher than the emitted frequency. This is due to the Doppler effect.

### Step 2: Write the Formula for Apparent Frequencies
For a stationary source and moving observer, the apparent frequency \( f' \) can be calculated using the formula:
\[ f' = f_0 \frac{v + v_o}{v - v_s} \]
where:
- \( f_0 \) = emitted frequency
- \( v \) = speed of sound (330 m/s)
- \( v_o \) = speed of the observer (car) towards the source
- \( v_s \) = speed of the source (0 for stationary source)

Since the source is stationary, we can simplify the formula for each car:
- For Car 1 (speed \( v_1 \)):
\[ f_1 = f_0 \frac{v + v_1}{v} \]
- For Car 2 (speed \( v_2 \)):
\[ f_2 = f_0 \frac{v + v_2}{v} \]

### Step 3: Find the Difference in Frequencies
The difference in frequencies reflected from the two cars is given by:
\[ f_2 - f_1 = f_0 \left( \frac{v + v_2}{v} - \frac{v + v_1}{v} \right) \]
This simplifies to:
\[ f_2 - f_1 = f_0 \left( \frac{v_2 - v_1}{v} \right) \]

### Step 4: Use the Given Information
We are given that the difference in frequencies is \( 1.2\% \) of \( f_0 \):
\[ f_2 - f_1 = 0.012 f_0 \]
Equating the two expressions for \( f_2 - f_1 \):
\[ f_0 \left( \frac{v_2 - v_1}{v} \right) = 0.012 f_0 \]
Cancelling \( f_0 \) from both sides (since \( f_0 \neq 0 \)):
\[ \frac{v_2 - v_1}{v} = 0.012 \]
Thus,
\[ v_2 - v_1 = 0.012 v \]

### Step 5: Substitute the Speed of Sound
Substituting \( v = 330 \, \text{m/s} \):
\[ v_2 - v_1 = 0.012 \times 330 = 3.96 \, \text{m/s} \]

### Step 6: Convert to km/h
To convert from m/s to km/h, we multiply by \( 3.6 \):
\[ v_2 - v_1 = 3.96 \times 3.6 \approx 14.256 \, \text{km/h} \]

### Step 7: Round to the Nearest Integer
Rounding \( 14.256 \) to the nearest integer gives:
\[ \text{Difference in speeds} \approx 14 \, \text{km/h} \]

### Final Answer
The difference in the speeds of the cars is approximately **14 km/h**.

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