A source of sound emitting a note of frequency 200 Hz moves towards an observer with a velocity v equal to the velocity of sound. If the observer also moves away from the source with the same velocity v, the apparent frequency heard by the observer is

Options

A. 50 Hz
B. 100 Hz
C. 150 Hz
D. 200 Hz

Explanation

To solve the problem of the apparent frequency heard by the observer when both the source of sound and the observer are moving, we can follow these steps:

### Step 1: Identify the given values
- Frequency of the source, \( f = 200 \, \text{Hz} \)
- Velocity of the sound, \( v \)
- Velocity of the source, \( v_s = v \) (moving towards the observer)
- Velocity of the observer, \( v_o = v \) (moving away from the source)

### Step 2: Understand the Doppler Effect
The Doppler Effect describes how the frequency of sound changes for an observer moving relative to the source of sound. The formula for the apparent frequency \( f' \) when both the source and observer are moving is given by:

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

Where:
- \( f' \) is the apparent frequency
- \( f \) is the actual frequency
- \( v \) is the speed of sound
- \( v_o \) is the speed of the observer (positive if moving towards the source)
- \( v_s \) is the speed of the source (positive if moving towards the observer)

### Step 3: Substitute the values into the formula
Since both the source and observer are moving with the same velocity \( v \), we can substitute \( v_s = v \) and \( v_o = -v \) (the observer is moving away from the source):

\[ f' = 200 \, \text{Hz} \cdot \frac{v - v}{v - v} \]

### Step 4: Simplify the equation
In this case, since \( v_s = v \) and \( v_o = -v \), we have:

\[ f' = 200 \, \text{Hz} \cdot \frac{v - v}{v - v} = 200 \, \text{Hz} \cdot \frac{0}{0} \]

This indicates that there is no relative motion between the source and the observer since they are moving at the same speed in opposite directions.

### Step 5: Conclusion
Since there is no relative motion between the source and the observer, the apparent frequency \( f' \) is equal to the actual frequency \( f \):

\[ f' = 200 \, \text{Hz} \]

Thus, the apparent frequency heard by the observer is **200 Hz**.