Question

A supersonic jet plane moves parallel to the ground at speed v = 0.75 mach (1 mach = speed of sound). The frequency of its engine sound is $v_o=2kHz$ and the height of the jet plane is h = 1.5 km. At some instant an observer on the ground hears a sound of frequency $v=2_o$. Find the instant (in sec) prior to the instant of hearing when the sound wave received by the observer was emitted by the jet plane. Velocity of sound wave in the condition of observer = 340 m/s.

Answer 1

Frequency of supersonic jet as observed on ground$=2{\nu }_0={\nu }_0\left(\frac{v_{sound}}{v_{sound}-v_{jet}cos\theta }\right)$

where $\theta$ is the angle the direction of sound traveled forms with the horizontal.

Thus $2=\frac{v_{sound}}{v_{sound}-\left(0.75v_{sound}cos\theta \right)}$

$⟹cos\theta =\frac{2}{3}$

The distance traveled the sound in the calculated direction =$l=\frac{h}{sin\theta }$

where $h$ is the height of the path of the jet.

$l=\frac{1500}{\frac{\sqrt{5}}{3}}$

Thus time taken=$\frac{l}{v_{sound}}=6s$