Question

A rocket is moving at a speed of $220{\text{ m}}{{\text{s}}^{ - 1}}$ towards a stationary target, emits a sound of frequency $1000Hz,$ some of the sound reaching the target gets reflected back to the rocket as echo. The frequency of the echo as detected by the rocket is (Take velocity of sound $ = 330{\text{ m}}{{\text{s}}^{ - 1}})$
$
A.\,3500Hz \\
B.\,4000Hz \\
C.\,5000Hz \\
D.\,3000Hz \\
$

Answer 1

Hint:Here use the concept of the Doppler effect, when the observer and the source moves towards each other or move far from each other, then there will be a change in their frequency.For example: as one approaches a blowing horn, the perceived pitch is higher until the horn is reached and then becomes lower as the horn is passed.

Formula used:
$f' = f\left( {\dfrac{{v + {v_R}}}{{v - {v_s}}}} \right)$
where f is the initial frequency, v is the velocity, ${v_R}$ is the velocity of the rocket and ${v_s}$ is the velocity of the sound.

Complete step by step answer:
Frequency of the sound is $f = 1000Hz$
Speed of the rocket is ${v_R} = 220m{s^{ - 1}}$
Speed of the rocket is equal to the speed of the sound.
Speed of the sound is ${v_s} = 220m{s^{ - 1}}$
Velocity of the sound is $v = 330m{s^{ - 1}}$
Let the frequency of the echo as detected by the rocket is $f'$
$f' = f\left( {\dfrac{{v + {v_R}}}{{v - {v_s}}}} \right)$
Place the given values in the above equation –
$
f' = 1000\left( {\dfrac{{330 + 220}}{{330 - 220}}} \right) \\
f' = 1000\left( {\dfrac{{550}}{{110}}} \right) \\
f' = 1000(5) \\
f' = 5000Hz \\
 $
The frequency of the echo detected by the rocket is $5000Hz$.
Hence, from the given multiple choices – the option C is the correct answer.

Note: Please know the difference between the Doppler effect and the Doppler shift. In the Doppler effect, there is the change in the observed frequency of a wave when the source and the observer moves relative to the motion whereas, in the Doppler shift there is the movement of source or observer with respect to the medium.