Question

A whistle of frequency 1000 Hz is sounded on a car travelling towards a cliff with velocity of \[18\,ms^{ - 1} \] normal to the cliff. If \[c = 330\,ms^{ - 1} \],what will be the apparent frequency of the echo as heard by the car driver?

Answer 1

Hint:Use Doppler formula for determining the apparent frequency when both observer and source are moving towards each other.
Formula used: \[\nu ' = \nu \left( {\dfrac{{v + v_0 }}{{v - v_s }}} \right)\]
Here, \[\nu '\] is the apparent frequency, \[\nu \] is the original frequency, \[v\]is the of sound, \[v_0 \]is the velocity of an observer and \[v_s \] is the velocity of the source.

Complete step by step answer:
In this question, we are asked to determine the apparent frequency of the echo heard by the car driver himself. Therefore, the cliff is now the source of sound as it creating an echo and the car driver is the observer of the echo.

Use Doppler formula to calculate the apparent frequency when both the source and observer are moving towards each other as follows,
\[\nu ' = \nu \left( {\dfrac{{v + v_0 }}{{v - v_s }}} \right)\]
Here, \[\nu '\] is the apparent frequency of an echo heard by the car driver, \[\nu \] is the frequency emitted by the echo, \[v\] is the velocity of sound, \[v_0 \]is the velocity of an observer (car driver) and \[v_s \]is the velocity of an echo of sound.
Here, the velocity of both observer and echo is the same as it is the same sound reflecting off from the cliff.
Substitute \[330\,ms^{ - 1} \] for \[v\], \[1000\,Hz\] for \[\nu \], \[18\,ms^{ - 1} \] for \[v_s \]
and \[18\,ms^{ - 1} \] for \[v_0 \] in the above equation.
\[\nu ' = \left( {1000\,Hz} \right)\left( {\dfrac{{330\,ms^{ - 1} + 18\,ms^{ - 1} }}{{330\,ms^{ - 1} - 18\,ms^{ - 1} }}} \right)\]
\[\nu ' = \left( {1000\,Hz} \right)\left( {\dfrac{{348\,ms^{ - 1} }}{{312\,ms^{ - 1} }}} \right)
\]
\[\nu ' = \left( {1000\,Hz} \right)\left( {1.1154} \right)\]
\[\nu ' = 1115\,Hz\]
So, the correct answer is option A).

Note:We can also solve this question by first calculating the apparent frequency of the whistle sounded by the car heard by the imaginary observer standing near the cliff. Then we have to determine the apparent frequency of the sound reflected from the cliff heard by the observer (car driver). To determine the apparent frequency of the whistle heard by the observer at cliff, we have to set \[v_o = 0\,ms^{ - 1} \], also, to calculate the frequency of echo, we have to set \[v_s = 0\,ms^{ - 1} \].