Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let ν(t) represent the beat frequency measured by a person sitting in the car at time t. Let νP, νQ and vR be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is 330 ms–1. Which of the following statement(s) is(are) true regarding the sound heard by the person?

Engineering

Physics

Dopplers Effect

Question

Two loudspeakers M and N are located 20 m apart and emit sound at frequencies 118 Hz and 121 Hz, respectively. A car is initially at a point P, 1800 m away from the midpoint Q of the line MN and moves towards Q constantly at 60 km/hr along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R, 1800 m away from Q. Let ν(t) represent the beat frequency measured by a person sitting in the car at time t. Let ν_P, ν_Q and v_R be the beat frequencies measured at locations P, Q and R, respectively. The speed of sound in air is 330 ms^–1. Which of the following statement(s) is(are) true regarding the sound heard by the person?

ν_P + ν_R = 2ν_Q

the plot below represents schematically the variation of beat frequency with time

The plot below represents schematically the variation of beat frequency with time

The rate of change in beat frequency is maximum when the car passes through Q

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$f_{1} = f_{01} (\frac{330 + vcosθ}{330})$

$f_{2} = f_{02} (\frac{330 + vcosθ}{330})$

$Δf = | f_{1} - f_{2} | = | f_{01} - f_{02} | (1 + \frac{v \cos θ}{330})$

$Δf = 3 (1 + kcos θ) k = \frac{v}{330} constant$