A person carrying a whistle emitting continuously a note of `272 Hz` is runnig towards a reflecting surface with a speed of `18km//h`. The speed of sound in air is `345ms^(-1)` The number of beats heard by him is

Options

A. `4`
B. `6`
C. `8`
D. `3`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Convert the speed of the person from km/h to m/s
The speed of the person (Vs) is given as 18 km/h. We need to convert this to meters per second (m/s).

\[ Vs = 18 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 5 \text{ m/s} \]

### Step 2: Calculate the apparent frequency (F') heard by the person
The formula for the apparent frequency when the source is moving towards a stationary observer is given by:

\[ F' = F \times \frac{V + V_o}{V - V_s} \]

Where:
- \( F \) = original frequency = 272 Hz
- \( V \) = speed of sound = 345 m/s
- \( V_o \) = speed of the observer (which is 0 since the observer is the person himself)
- \( V_s \) = speed of the source = 5 m/s

Substituting the values into the formula:

\[ F' = 272 \times \frac{345 + 0}{345 - 5} = 272 \times \frac{345}{340} \]

### Step 3: Calculate the value of F'
Now we can calculate \( F' \):

\[ F' = 272 \times \frac{345}{340} \approx 272 \times 1.0147 \approx 276.0 \text{ Hz} \]

### Step 4: Calculate the frequency of the reflected sound
The reflected sound will have the same frequency as the apparent frequency calculated above, but now the person is acting as an observer for the reflected sound. The formula for the frequency heard by the person when the sound reflects off a surface is:

\[ F'' = F' \times \frac{V + V_o}{V - V_s} \]

Here, \( V_o \) is now the speed of the person (5 m/s) and \( V_s \) is 0 since the reflecting surface is stationary.

Substituting the values:

\[ F'' = F' \times \frac{345 + 5}{345 - 0} = F' \times \frac{350}{345} \]

### Step 5: Substitute \( F' \) into the equation
Using the value of \( F' \):

\[ F'' = 276 \times \frac{350}{345} \approx 276 \times 1.0145 \approx 280 \text{ Hz} \]

### Step 6: Calculate the number of beats
The number of beats heard is the difference between the original frequency and the frequency of the reflected sound:

\[ \text{Number of beats} = |F'' - F| = |280 - 272| = 8 \text{ beats per second} \]

### Final Answer
The number of beats heard by the person is **8 beats per second**.

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