A speeding motorcyclist sees traffic jam ahead of him. He slows down to `36 km//h` He finds that traffic has eased and a car moving ahead of him at `18 km//h` is honking at a frequency of 1392 Hz. If the speed of sound is `343m//s`, the frequency of the honk as heard by him will be
(a) 1332 Hz (b) 1372 Hz (c) 1412 Hz (d) 1454 Hz

Options

A. 1332 Hz
B. 1372 Hz
C. 1412 Hz
D. 1454 Hz

Answer

To solve the problem, we will use the Doppler Effect formula to find the frequency of the honk as heard by the motorcyclist. Here are the steps:

### Step 1: Convert speeds from km/h to m/s
- The speed of the motorcyclist is given as 36 km/h.
- To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).

\[ \text{Speed of motorcyclist} = 36 \times \frac{5}{18} = 10 \, \text{m/s} \]

- The speed of the car is given as 18 km/h. We will convert it similarly.

\[ \text{Speed of car} = 18 \times \frac{5}{18} = 5 \, \text{m/s} \]

### Step 2: Identify the known values
- Frequency of the honk \( f = 1392 \, \text{Hz} \)
- Speed of sound \( v = 343 \, \text{m/s} \)
- Speed of the observer (motorcyclist) \( v_o = 10 \, \text{m/s} \)
- Speed of the source (car) \( v_s = 5 \, \text{m/s} \)

### Step 3: Apply the Doppler Effect formula
The formula for the apparent frequency \( f' \) as heard by the observer is given by:

\[ f' = f \times \frac{v + v_o}{v - v_s} \]

Here:
- \( f' \) is the frequency heard by the observer (motorcyclist).
- \( f \) is the actual frequency of the source (car).
- \( v \) is the speed of sound.
- \( v_o \) is the speed of the observer (motorcyclist).
- \( v_s \) is the speed of the source (car).

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:

\[ f' = 1392 \times \frac{343 + 10}{343 - 5} \]

### Step 5: Calculate the values
Calculate the numerator and denominator:

- Numerator: \( 343 + 10 = 353 \)
- Denominator: \( 343 - 5 = 338 \)

Now substitute these values back into the equation:

\[ f' = 1392 \times \frac{353}{338} \]

### Step 6: Perform the multiplication and division
Calculating the fraction:

\[ \frac{353}{338} \approx 1.041 \]

Now multiply this by 1392:

\[ f' \approx 1392 \times 1.041 \approx 1454.032 \, \text{Hz} \]

### Step 7: Round to the nearest whole number
The final frequency as heard by the motorcyclist is approximately:

\[ f' \approx 1454 \, \text{Hz} \]

### Conclusion
Thus, the frequency of the honk as heard by the motorcyclist is **1454 Hz**, which corresponds to option (d).

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