Question

The pieman now walks with speed $30m{\min ^{ - 1}}$ towards the receiver while putting pies on the belt at $20$ per minute. Find the spacing of the frequency with which they are received by the stationary pie eater.
\left( A \right)13.5{m^{ - 1}},22.22{\min ^{ - 1}} \\\ \left( B \right)15{m^{ - 1}},22.22{\min ^{ - 1}} \\\ \left( C \right)135{m^{ - 1}},22.22{\min ^{ - 1}} \\\ \left( D \right)3{m^{ - 1}},22.22{\min ^{ - 1}} \\\

Answer 1

Hint : In this question, we are provided with the velocity of the source, the original frequency and the speed of the waves, so, by using the concept of Doppler Effect and the formulae for the spacing and the frequency received by the receiver, we can solve this question.
The frequency of wave received is given by the relation
${f_0} = \dfrac{{v \pm {v_0}}}{{v \pm {v_s}}}{f_s}$
${f_0}$ Is observer frequency, $v$ is speed of sound waves, ${v_0}$ is observer velocity, ${v_s}$ is source velocity, ${f_s}$ and is actual frequency of sound waves.
The spacing of the frequency can be calculated by the formula
$x = \dfrac{{v - {v_s}}}{f}$

Complete Step By Step Answer:
Doppler Effect refers to the change in the wave frequency received by the receiver due to the relative motion between the wave source and the observer. It is represented by the relation
${f_0} = \dfrac{{v \pm {v_0}}}{{v \pm {v_s}}}{f_s}$
In this case, the velocity of the pieman is $30m{\min ^{ - 1}}$ towards the receiver so in the denominator, we will take $v - {v_s}$ and the frequency of the source is $20$ per minute, the spacing of the frequency can be calculated by the formula
$x = \dfrac{{v - {v_s}}}{f}$
Here, $f = 20{\min ^{ - 1}}$
The conveyor belt is moving towards the right with speed $v = 300m{\min ^{ - 1}}$
${v_s} = 30m{\min ^{ - 1}}$
Therefore, $x = \dfrac{{300 - 30}}{{20}} = \dfrac{{270}}{{20}} = 13.5m$
So, the frequency with which the pies are received by the pie eater are calculated as:
$f' = \dfrac{v}{{v - {v_s}}}f$
Thus, the final frequency becomes
$f' = \dfrac{{300}}{{300 - 30}} \times 20 = \dfrac{{300}}{{270}} \times 20 = 22.22{\min ^{ - 1}}$
Hence, the option $\left( A \right)13.5{m^{ - 1}},22.22{\min ^{ - 1}}$ is the correct answer.

Note :
It is very important to note the speed with which the waves are being produced, here the source of production of the pies is the conveyor belt and the cause of change in the velocity with which pies are reaching is the pieman, that’s why these speeds affect the frequency of the wave received by the observer.