Question

A rocket is moving at a speed of $200\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$ towards a stationary target. While moving, it emits a wave of frequency $1000\;{\text{Hz}}$ Calculate the frequency of the sound as detected by the target. (Velocity of sound in air is $330\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$)

Answer 1

Hint:Doppler effect used to find the frequency of the sound detected from the target. The Doppler effect or Doppler shift is used to establish a relation between the observer's velocity towards the source and source velocity towards the observer.

Useful formula:
The expression for finding the frequency of sound wave detected from the target is
$f = {f_0}\left( {\dfrac{{{v_{sound}}}}{{{v_{sound}} - {v_s}}}} \right)$
Where ${v_s}$is the speed of the rocket, ${v_{sound}}$is the velocity of sound in air and ${f_0}$is the original sound frequency.

Complete step by step answer:
Given, The speed of the rocket is ${v_s} = 200\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$
Original sound frequency is ${f_o} = 1000\;{\text{Hz}}$
Velocity of sound in air is ${V_{sound}} = 330\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}$
we need to find the frequency of sound as detected by the target.

The expression for finding the frequency of sound wave detected from the target is

$f = {f_0}\left( {\dfrac{{{v_{sound}}}}{{{v_{sound}} - {v_s}}}} \right)$
Substitute all the value in the above equation
$\
  f = 1000\;{\text{Hz}}\left( {\dfrac{{330\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}}}{{330\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}} - 200\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}}}} \right) \\
  f = 1000\;{\text{Hz}}\left( {\dfrac{{330\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}}}{{130\;{\text{m}}{{\text{s}}^{{\text{ - 1}}}}}}} \right) \\
  f = 1000\;{\text{Hz}} \times 2.54\; \\
  f = 2540\;{\text{Hz}} \\
 $

Thus, the frequency of sound wave detected from target is $f = 2540\;{\text{Hz}}$

Additional information:
Doppler effect states, an increase or decrease frequency of light is obtained as the source and observer move toward or away from each other. The combined equation is used when both are moving relative to one and another. The application of Doppler effects is- radar, vibration measurement, medical imaging, etc.

Note:
By using the data of frequency of sound wave detected from the target, velocity of the sound and velocity of the rocket the frequency of echo as detected by the rocket can be calculated by using the following formula.
${f_{echo}} = f\left( {\dfrac{{{v_{sound}} + {v_s}}}{{{v_{sound}}}}} \right)$
Where ${v_s}$is the speed of the rocket, ${v_{sound}}$is the velocity of sound in air and $f$is the frequency of sound wave detected from target.