A bus is moving with a velocity of `5 ms^(-1)` towards a huge wall. The driver sound a horn of frequency 165 Hz. If the speed of sound in air is `335 ms^(-1)`, the number of beats heard per second by a passenger inside the bus will be

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we will use the Doppler effect formula to find the frequency of the sound heard by the passenger in the bus. Here's the step-by-step solution:

### Step 1: Identify the given values
- Frequency of the horn (f) = 165 Hz
- Speed of sound in air (v) = 335 m/s
- Velocity of the bus (vb) = 5 m/s

### Step 2: Use the Doppler effect formula
Since the bus is moving towards a stationary wall, we can use the Doppler effect formula for the source moving towards a stationary observer. The formula for the observed frequency (f') when the source is moving towards the observer is given by:

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

Where:
- \( f' \) = observed frequency
- \( f \) = source frequency (165 Hz)
- \( v \) = speed of sound (335 m/s)
- \( v_o \) = speed of observer (0 m/s, since the passenger is inside the bus)
- \( v_s \) = speed of source (5 m/s, since the bus is moving towards the wall)

### Step 3: Substitute the values into the formula
Since the observer is stationary inside the bus, we have \( v_o = 0 \). Thus, the formula simplifies to:

\[ f' = 165 \left( \frac{335 + 0}{335 - 5} \right) \]

Calculating the denominator:

\[ 335 - 5 = 330 \]

Now substituting back into the equation:

\[ f' = 165 \left( \frac{335}{330} \right) \]

### Step 4: Calculate the observed frequency
Now we can calculate \( f' \):

\[ f' = 165 \times \frac{335}{330} \]

Calculating the fraction:

\[ \frac{335}{330} \approx 1.01515 \]

Now multiply:

\[ f' \approx 165 \times 1.01515 \approx 167.5 \text{ Hz} \]

### Step 5: Calculate the number of beats per second
The number of beats per second is given by the difference between the observed frequency and the original frequency:

\[ \text{Beats} = |f' - f| = |167.5 - 165| = 2.5 \text{ Hz} \]

### Final Answer
The number of beats heard per second by a passenger inside the bus is approximately **2.5 beats per second**.