Question

A car P travelling at 20 \(ms^{–1}\) sounds its horn at a frequency of 400 Hz. Another car Q is travelling being the first car in the same direction with a velocity 40 \(ms^{–1}\) . The frequency heard by the passenger of the car Q is approximately [Take, velocity of sound = 360 \(ms^{–1}\) ]

• A. 485 Hz
• B. 421 Hz
• C. 471 Hz
• D. 514 Hz

Answer 1

The Correct Option is B

Given: 

• Velocity of car \( P \) (\( V_P \)) = \( 20 \, \text{m/s} \)
• Velocity of car \( Q \) (\( V_Q \)) = \( 40 \, \text{m/s} \)
• Frequency of the horn (\( f \)) = \( 400 \, \text{Hz} \)
• Velocity of sound (\( V_s \)) = \( 360 \, \text{m/s} \)

Step 1: Apply the Doppler Effect Formula

The apparent frequency (\( f_{\text{app}} \)) heard by the passenger in car \( Q \), when both cars are moving in the same direction, is given by:

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

Here:

• \( V_Q \): Velocity of the observer (car \( Q \))
• \( V_P \): Velocity of the source (car \( P \))
• \( V_s \): Velocity of sound
• \( f \): Frequency of the source

 

Step 2: Substitute the Given Values

Substitute \( V_s = 360 \, \text{m/s} \), \( V_Q = 40 \, \text{m/s} \), \( V_P = 20 \, \text{m/s} \), and \( f = 400 \, \text{Hz} \):

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

Simplify the expression:

\[ f_{\text{app}} = \frac{400}{380} \cdot 400. \]

Calculate \( f_{\text{app}} \):

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

Final Answer:

The frequency heard by the passenger of car \( Q \) is approximately \( 421 \, \text{Hz} \).