A car P travelling at 20 m/s sounds its horn at a frequency of 40… — Solution | JEE Physics

MCQEasyJEE 2023Doppler Effect

A car P travelling at $20 \, \text{m/s}$ sounds its horn at a frequency of $400 \, \text{Hz}$ . Another car Q is travelling behind the first car in the same direction with a velocity $40 \, \text{m/s}$ . The frequency heard by the passenger of the car Q is approximately [Take, velocity of sound $= 360 \, \text{m/s}$ ]

A
$471 \, \text{Hz}$
B
$514 \, \text{Hz}$
C
$421 \, \text{Hz}$
D
$485 \, \text{Hz}$

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given:

Velocity of car $P$ is $20 \, \text{m/s}$
Velocity of car $Q$ is $40 \, \text{m/s}$
Frequency of the horn is $400 \, \text{Hz}$
Velocity of sound is $360 \, \text{m/s}$

Find: The apparent frequency heard by the passenger in car $Q$ .

Use the Doppler effect formula for source and observer moving in the same direction:

f_{\text{app}} = \frac{V_s + V_Q}{V_s + V_P} \cdot f

Substituting the given values:

f_{\text{app}} = \frac{360 + 40}{360 + 20} \cdot 400

f_{\text{app}} = \frac{400}{380} \cdot 400

f_{\text{app}} = \frac{400 \times 400}{380} \approx 421 \, \text{Hz}

Therefore, the frequency heard by the passenger of car $Q$ is approximately $421 \, \text{Hz}$ . The solution working gives $421 \, \text{Hz}$ , which matches option C, although the solution incorrectly labels the correct option as B.

Common mistakes

Using the option label shown on the solution without checking the calculation. The working clearly gives $421 \, \text{Hz}$ , so the correct choice is the option containing that value. Always trust the derived numerical result over a mislabeled option letter.
Applying the wrong Doppler sign convention. Since car $Q$ is moving toward the sound coming from the car ahead and the source car $P$ is moving away from the observer behind it, the numerator and denominator must be chosen consistently with the direction of propagation. Draw the relative motion before substituting signs.
Using relative speed directly in place of the Doppler formula. The Doppler effect depends on observer speed, source speed, and sound speed separately, not only on the difference $40 - 20$ . Use the standard formula with the correct sign convention.

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Step-by-step solution

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