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 perpedicular 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. (a) `v_(P) + v_(R) = 2v_(Q)`
B. (b) The rate of changes in beat frequency is maximum when the car passes through `Q`
C. ( c ) The plot below represents schematically the variation of beat frequency with time

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

Answer

A, B, C

Explanation

(a,b,c)

`v_(P) = 121 - 118 = [(v + v cos theta)/(v)]`
`v_(theta) = 121 - 118 = 3`
`v_(R) = (121 - 118)[(v - v_(c) cos theta)/(v)]`
:. `v_(P) + v_(R) = 2v_(Q)` rArr `(A)` is correct option
In general when the car is passing through `A`
`v = 3[(v + v_(c) cos alpha)/(v)]` ...(i)
:. `(dv)/(dalpha) = -3[( v_(c) sin alpha)/(v)] |(dv)/(dalpha)|` is max when `sin alpha = 1`
i.e., `alpha = 90^(@)` (at Q)
rArr `(b)` is correct option.
From (i) `(dv)/(dt) = (3v_(c))/(v)(-sin alpha)(dalpha)/(dt) ...(i)
Also `tan alpha = (10)/(x)` :. `sec^(2)alpha(dalpha)/(dt) = -(10)/(x^(2))(dx)/(dt)`
:. `(dalpha)/(dt) = (-10v)/(x^(2)sec^(2)alpha)`
From (ii)& (iii)
`(dv)/(dt) = -(3v_(c))/(v)sin alpha xx ((-10v)/(x^(2)sec^(2)alpha)) = (30V-(c)sin alpha)/(x^(2)sec^(2)alpha)`
`(dv)/(dt) = (30v_(c ) sin alpha)/((10cot alpha)^(2) sec^(2)alpha) = 0.3v_(c) sin^(3) alpha .At `alpha = 90^(@)`
`(dv)/(dt) = max`
:. `( c )` is the correct option