JEE Advanced 2016 · Paper 1 · Physics Q7

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PhysicsMultiple correct+4 / -2Question 7 of 54

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 $\nu(t)$ represent the beat frequency measured by a person sitting in the car at time $t$ . Let $\nu_P$ , $\nu_Q$ and $\nu_R$ be the beat frequencies measured at locations $P$ , $Q$ and $R$ , respectively. The speed of sound in air is $330\ \mathrm{m\,s^{-1}}$ . Which of the following statement(s) is(are) true regarding the sound heard by the person?

\nu_P+\nu_R=2\nu_Q

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

Q

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

ν(t) against t: starts flat at a high value at P, falls through ν_Q (dotted level) steeply at Q in a reversed-S shape, and levels off at a low value at R.

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

ν(t) against t: starts high at P and falls steeply, flattens near ν_Q (dotted level) around Q, then falls steeply again towards R.

Answer and worked solution

Official answer: A, B, C

Doppler shifts. On the bisector, the car's velocity makes the same angle $\theta$ with the line to each speaker. The component towards each is $v\cos\theta$ , so both frequencies scale by the same factor:
$\nu=(121-118)\left(1\pm\frac{v\cos\theta}c\right).$
Use + while approaching (P side) and − after passing Q. At Q, $\cos\theta=0$ , so $\nu_Q=3$ Hz.

(A) $\nu_P+\nu_R=3(1+k)+3(1-k)=6=2\nu_Q$ . True.
(B) $\cos\theta=\dfrac x{\sqrt{x^2+10^2}}$ changes fastest at $x=0$ , so the beat frequency changes fastest at Q. True.
(C) Far away $\cos\theta\approx\pm1$ and $\nu$ is nearly flat. It drops steeply through $\nu_Q$ at Q and levels off again: plot (C). True.
(D) The plot shows the slowest change at Q. False.

(A), (B), (C)

Topics:Beats Doppler effect

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JEE Advanced 2016 · Paper 1, Physics question 7