a bus is moving towards and stationary observer with speed 20 m/s blows a horn of frequency 200 Hz. the frequency of sound heard by the observer is approximately ( take speed of sound 320 m/s)

Options

A. 188 Hz
B. 200 Hz
C. 226 Hz
D. 213 Hz

Answer

To solve the problem of finding the frequency of sound heard by a stationary observer when a bus is moving towards them, we can use the Doppler effect formula. Here’s the step-by-step solution:

### Step 1: Identify the given values
- Frequency of the sound emitted by the bus (source), \( f_0 = 200 \, \text{Hz} \)
- Speed of the bus (source), \( v_s = 20 \, \text{m/s} \)
- Speed of sound in air, \( v = 320 \, \text{m/s} \)
- Speed of the observer, \( v_o = 0 \, \text{m/s} \) (since the observer is stationary)

### Step 2: Write the formula for the apparent frequency
The formula for the apparent frequency \( f \) heard by the observer when the source is moving towards a stationary observer is given by:

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

### Step 3: Substitute the values into the formula
Since the observer is stationary, \( v_o = 0 \). Therefore, the formula simplifies to:

\[ f = f_0 \times \frac{v}{v - v_s} \]

Substituting the known values:

\[ f = 200 \, \text{Hz} \times \frac{320 \, \text{m/s}}{320 \, \text{m/s} - 20 \, \text{m/s}} \]

### Step 4: Calculate the denominator
Calculate \( 320 \, \text{m/s} - 20 \, \text{m/s} \):

\[ 320 \, \text{m/s} - 20 \, \text{m/s} = 300 \, \text{m/s} \]

### Step 5: Substitute back into the formula
Now substitute back into the formula:

\[ f = 200 \, \text{Hz} \times \frac{320 \, \text{m/s}}{300 \, \text{m/s}} \]

### Step 6: Simplify the fraction
Now simplify the fraction:

\[ f = 200 \, \text{Hz} \times \frac{32}{30} \]

### Step 7: Calculate the apparent frequency
Now calculate the frequency:

\[ f = 200 \, \text{Hz} \times \frac{32}{30} = 200 \, \text{Hz} \times 1.0667 \approx 213.33 \, \text{Hz} \]

### Step 8: Round to the nearest whole number
The approximate frequency heard by the observer is:

\[ f \approx 213 \, \text{Hz} \]

### Final Answer
The frequency of sound heard by the observer is approximately **213 Hz**.

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