Waves & Sound — JEE Advanced · 2016 · 22 May 2016 · Paper 1 Question with Solution

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

M N

and moves towards

Q

constantly at 60 km/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 the 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 m 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
(D) The plot below represents schematically the variation of beat frequency with time

(A) $v_{P} + v_{R} = 2 v_{Q}$ ✓ Correct
(B)The rate of change in beat frequency is maximum when the car passes through $Q$ ✓ Correct
(C)(see structure)
✓ Correct
(D)(see structure)

Answer: (A), (B), (C)

Solution

Due to the Doppler effect, the observed frequencies from

M

and

N

change as the car moves. At position

P

R

, the component of velocity toward each speaker is

v_{D} cos α

, where

α

is the angle from the perpendicular bisector.

At P:

v_{P} = (121 - 118) (\frac{v + v _{D} c o s α}{v}) = 3 (\frac{v + v _{D} c o s α}{v})

Hz

At Q:

v_{Q} = 121 - 118 = 3

Hz (no Doppler shift perpendicular to motion)

At R:

v_{R} = 3 (\frac{v - v _{D} c o s α}{v})

Hz

Therefore

v_{P} + v_{R} = 2 v_{Q}

. (A) is correct.

The rate of change

\frac{d v}{d θ} \propto - sin θ

, which is maximum at

Q

where

θ = 90°

. (B) is correct.

The beat frequency increases from

P

Q

and then decreases symmetrically to

R

, matching graph (C). (C) is correct.

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · One or more correct options

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Waves & Sound

Solution

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