A motor car is approaching towards a crossing with a velocity of `72 " km h"^(-1)` . The frequency of the sound of its horn as heard by a policeman standing on the crossing is 260 Hz. The frequency of horn is

Options

A. 200 Hz
B. 244 Hz
C. 150 Hz
D. 80 Hz

Answer

To solve the problem, we will use the Doppler effect formula for sound when the source is moving towards a stationary observer. The formula for the apparent frequency (f') heard by the observer is given by:

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

Where:
- \( f' \) = apparent frequency (frequency heard by the observer)
- \( f_0 \) = actual frequency of the source (horn frequency)
- \( v \) = speed of sound in air
- \( v_s \) = speed of the source (car)

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as \( 72 \, \text{km/h} \). We convert this to meters per second using the conversion factor \( \frac{5}{18} \):

\[ v_s = 72 \times \frac{5}{18} = 20 \, \text{m/s} \]

### Step 2: Identify the speed of sound
The speed of sound in air is typically taken as \( v = 332 \, \text{m/s} \).

### Step 3: Use the Doppler effect formula
We know the apparent frequency \( f' = 260 \, \text{Hz} \). We can now substitute the known values into the Doppler effect formula:

\[ 260 = f_0 \frac{332}{332 - 20} \]

### Step 4: Simplify the equation
Calculate the denominator:

\[ 332 - 20 = 312 \]

Now substitute back into the equation:

\[ 260 = f_0 \frac{332}{312} \]

### Step 5: Solve for the actual frequency \( f_0 \)
Rearranging the equation gives:

\[ f_0 = 260 \times \frac{312}{332} \]

### Step 6: Calculate \( f_0 \)
Now we can calculate \( f_0 \):

\[ f_0 = 260 \times \frac{312}{332} \approx 240.24 \, \text{Hz} \]

### Conclusion
Thus, the actual frequency of the horn is approximately \( 240 \, \text{Hz} \).