The transverse displacement y(x,t) of a wave on a string is given by y(x,t) = e ^{}

Question

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/h along the perpendicular bisector of MN. It crosses Q and eventually reaches a point R 1800 m away from Q. Let v(t) represent the best 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 ms^-1. Which of the following statement(s) is (are) true regarding the sound heard by the person:

(a) v_P + v_R = 2v_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

Answer 1

Correct option is : (a) v_P + v_R = 2v_Q

Apparent frequency of M and N at P

Apparent frequency of M, N at R

\(f _\overset{"}{M} = (\frac{V - V_0}{V})\ f_M\)

\(\sec^2\theta\frac{d\theta}{dt} = -\frac{10}{x^2}\frac{dx}{dt}\)

\(\frac{d\theta}{dt} = \frac{10v}{x^2\sec ^2\theta}\) 

\(x = 10\ cot\theta\)

\(\frac{dv}{dt} = (-\frac{v}{110})\sin\theta(\frac{10v}{x^2\sec^2\theta})\)

\(|\frac{dv}{dt}| \propto\sin^3\theta\)

(c) \(|\frac{dv}{dt}|\ \) is maximum at Q.