Two loudspeaker `M` and `N` are located `20m` apart and emit sound at frequencies `118Hz` and `121Hz`, respectively. A car is intially at a point `P`, `1800m` 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`, `1800m` 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 `330ms^(-1)`. Which of the following statement (s) is (are) true regarding the sound heard by the person?

Options

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

B. `V_(P) + V_(R) = 2 V_(Q)`
C. The plot below represents schematically the variation of beat frequency with time

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

Answer

A, B, D

Explanation

Let u be the speed of observer moving alon PQR.
Frequendy received at P : `f = f_(0) (upsilon + u cos theta)/(upsilon)`
Here `upsilon` is speed of sound.
Beat frequency
`V_(P) = f_(1) - f_(2) (upsilon + u cos theta)/(upsilon) (f_(01) - f_(02))`
`= (v + u cos theta)/(upsilon) (121-118)`
`rArr" "V_(P) = (3)/(v) (v + u cos theta)`
As the time passes value of `theta` increases to `90^(@)`
Beat frequency at point Q :
`rArr" "V_(Q) = 121 - 118 = 3`
Beat frequency at R :
`rArr" "V_(R) = (3)/(upsilon) (upsilon - u cos theta)`
We can see that graph shown in option A is correct.
`V_(P) + V_(R) = (3)/(upsilon) (v + u cos theta) + (3)/(upsilon) (v - u cos theta) = 6`
Hence, we can see that `V_(P) + V_(R) = 2V_(Q)`
Hence, option (b) is correct.
Further by differentiating V, we can see that rate of beat frequency is maximum when observer crosses point Q hence, option (d) is also correct.