Question

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed of $1.0m{s}^{-1}$ and the man behind walks at a speed of $2.0m{s}^{-1}$. A third man is standing at a height $12m$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430Hz$. The speed of sound in air is $330m{s}^{-1}$. At the instant, when the moving men are $10m$ apart, the stationary man is equidistant from them. The frequency of beats in $Hz$. heard by the stationary man at this instant, is ______

Answer 1

Apparent frequency of A
$f_A=1430\left[\frac{330}{330-2\cos \theta }\right]=1430\left[\frac{1}{1-\frac{2\cos \theta }{330}}\right]$$=1430[1+\frac{2\cos \theta }{330}]$

(By bionomial expansion)
Apparent frequency of B
$f_B=1430\left[\frac{330}{330+1\cos \theta }\right]$$=1430[1-\frac{\cos \theta }{330}]$

Then, the beat frequency of the system is given by,
$\Delta f=f_A-f_B=1430\left[\frac{3\cos \theta }{330}\right]$$=13\cos \theta$
$=13\left(\frac{5}{13}\right)=5.00Hz$