Two loudspeakers M and N are located 20 ~m apart and emit sound a… - Vidyadip

MCQ

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

($A$) $v_P+v_R=2 v_Q$

($B$) The rate of change in beat frequency is maximum when the car passes through $Q$

($C$) The plot below represents schematically the variation of beat frequency with time

(image)

($D$) The plot below represents schematically the variation of beat frequency with time

(image)

✓
$A,B,C$
B
$A,B,D$
C
$C,B,D$
D
$B,C$

✓

Answer

Correct option: A.

$A,B,C$

a
Frequency of $M$ received by car

$f _1=118\left(\frac{V+V_0 \cos \theta}{V}\right)$

$f _2=121\left(\frac{V+V_0 \cos \theta}{V}\right)$

$Image$

No. of beats $n=\Delta f=f_2-f_1$

$n=3\left(\frac{V+V_0 \cos \theta}{V}\right)$

$n=3\left(1+\frac{V_0}{V} \cos \theta\right)$

As $\theta \uparrow, \cos \theta \downarrow, n \perp$

Rate of change of beat frequency

$\frac{d n}{d \theta}=3\left[\frac{V_0}{V}(-\sin \theta)\right]$

$\frac{d n}{d \theta}$ is maximum when $\sin \theta=1 ; \theta=90^{\circ}$ $i.e.$ car is at point $Q$ .

$v_p=3\left(1+\frac{v_0}{V} \cos \theta\right)$

$v_R=3\left(1-\frac{v_0}{V} \cos \theta\right)$

At $Q$

No. of beats $v_Q=121-118=3$

$v_Q=\frac{v_P+v_R}{2}$

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