A vehicle sounding a whistle of frequency 256Hz is moving on a straight road,towards a hill with velocity of `10ms^-1`. The number of beats per second observed by a person travelling in the vehicle is (velocity of sound = `300ms^-1`

Options

A. 0
B. 10
C. 14
D. 16

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Frequency of the whistle (f₀) = 256 Hz
- Velocity of the vehicle (v₀) = 10 m/s
- Velocity of sound (v) = 300 m/s

### Step 2: Calculate the apparent frequency (f') heard by the hill
Since the vehicle is moving towards the hill, we can use the formula for the apparent frequency when the source is moving towards a stationary observer:

\[ f' = f₀ \cdot \frac{v}{v - v₀} \]

Substituting the values:

\[ f' = 256 \cdot \frac{300}{300 - 10} = 256 \cdot \frac{300}{290} \]

Calculating the right side:

\[ f' = 256 \cdot \frac{300}{290} \approx 256 \cdot 1.0345 \approx 264.89 \text{ Hz} \]

### Step 3: Calculate the frequency of the echo (f'')
Now, the sound reflects back to the vehicle, which is also moving towards the source of the sound (the hill). We can use the formula for the apparent frequency when the observer is moving towards a stationary source:

\[ f'' = f' \cdot \frac{v + v₀}{v} \]

Substituting the values:

\[ f'' = 264.89 \cdot \frac{300 + 10}{300} = 264.89 \cdot \frac{310}{300} \]

Calculating the right side:

\[ f'' = 264.89 \cdot 1.0333 \approx 274.44 \text{ Hz} \]

### Step 4: Calculate the number of beats per second
The number of beats per second (n) is given by the difference between the frequency of the echo (f'') and the original frequency (f₀):

\[ n = f'' - f₀ \]

Substituting the values:

\[ n = 274.44 - 256 = 18.44 \text{ Hz} \]

### Step 5: Round to the nearest whole number
Since the number of beats is typically expressed as a whole number, we round 18.44 to 18.

### Final Answer
The number of beats per second observed by a person traveling in the vehicle is approximately **18 beats per second**.

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