Question

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0\,ms^{-1}$ and the man behind walks at a speed $2.0 \,ms^{-1}$. A third man is standing at a height $12\, m$ 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 $1430\, Hz$. The speed of sound in air is $330 \,ms^{-1}$. At the instant, when the moving men are $10\, m$ 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

$f_{A}=1430\left[\frac{330}{330-2 \cos \theta}\right]$
$=1430\left[\frac{1}{1-\frac{2 \cos \theta}{330}}\right]$
$=1430\left[1+\frac{2 \cos \theta}{330}\right]$
$f_{B}=1430\left[\frac{330}{330+\cos \theta}\right]$
$=1430\left[1-\frac{\cos \theta}{330}\right]$
$\Delta f=f_{A}-f_{B}=1430\left[\frac{3 \cos \theta}{330}\right]$
$=13 \cos \theta=13\left(\frac{5}{13}\right)=5.00\, Hz$