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 apply the Doppler Effect formula for sound. The formula for the apparent frequency (ν') heard by an observer when the source is moving towards the observer is given by:

\[ \nu' = \nu \frac{v - v_o}{v - v_s} \]

Where:
- \( \nu' \) = apparent frequency (frequency heard by the observer) = 260 Hz
- \( \nu \) = actual frequency (frequency of the horn, which we need to find)
- \( v \) = speed of sound in air (approximately 332 m/s)
- \( v_o \) = speed of the observer (0 m/s, since the policeman is at rest)
- \( v_s \) = speed of the source (the car)

### Step 1: Convert the speed of the car from km/h to m/s
The speed of the car is given as 72 km/h. To convert this to m/s, we use the conversion factor:

\[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]

Thus,

\[ v_s = 72 \text{ km/h} \times \frac{1}{3.6} = 20 \text{ m/s} \]

### Step 2: Substitute the known values into the Doppler Effect formula
Now we can substitute the known values into the Doppler Effect formula:

\[ 260 = \nu \frac{332 - 0}{332 - 20} \]

### Step 3: Simplify the equation
This simplifies to:

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

### Step 4: Solve for the actual frequency (ν)
To find the actual frequency (ν), we rearrange the equation:

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

### Step 5: Calculate the actual frequency
Now we can calculate ν:

\[ \nu = 260 \times \frac{312}{332} \approx 244 \text{ Hz} \]

### Conclusion
The actual frequency of the horn is approximately **244 Hz**.

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