A speed motorcyclist sees a traffic jam ahead of him. He slows down to $36 ext{ km/h}$. He finds that traffi

Question

A speed motorcyclist sees a traffic jam ahead of him. He slows down to $36 	ext{ km/h}$. He finds that traffic has eased and a car moving in front of him at $18 	ext{ km/h}$ is honking at a frequency of $1392 	ext{ Hz}$. If the speed of sound is $343 	ext{ m/s}$, the frequency of the honk as heard by him will be

Accepted Answer

1. **Identify the Given Data:**
   Velocity of the observer (motorcyclist), $v_o = 36 	ext{ km/h} = 36 	imes \frac{5}{18} 	ext{ m/s} = 10 	ext{ m/s}$.
   Velocity of the source (car), $v_s = 18 	ext{ km/h} = 18 	imes \frac{5}{18} 	ext{ m/s} = 5 	ext{ m/s}$.
   Actual frequency of the honk, $f = 1392 	ext{ Hz}$.
   Speed of sound, $v = 343 	ext{ m/s}$.
2. **Determine the Relative Motions:**
   The motorcyclist is behind the car. Both are moving in the same direction. 
   - The observer (motorcyclist) is moving *towards* the source, which tends to increase the apparent frequency. Thus, the numerator in the Doppler formula will be $(v + v_o)$.
   - The source (car) is moving *away* from the observer, which tends to decrease the apparent frequency. Thus, the denominator will be $(v + v_s)$.
3. **Apply the Doppler Effect Formula:**
   The apparent frequency $f'$ is given by:
   $$f' = f \left( \frac{v + v_o}{v + v_s} ight)$$
   Substitute the values into the formula :
   $$f' = 1392 \left( \frac{343 + 10}{343 + 5} ight)$$
   $$f' = 1392 \left( \frac{353}{348} ight)$$
   $$f' = 4 	imes 353 = 1412 	ext{ Hz}$$
Therefore, the frequency of the honk heard by the motorcyclist is $1412 	ext{ Hz}$.

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