Question

On the same path, the source and observer are moving such a way that the distance between these two increases with the time. The speeds of source and observer are the same and equal to $10 m{s}^{-1}$ with respect to the ground while no wind is blowing. The apparent frequency received by the observer is 1950 Hz, then the original frequency must be? (the speed of sound in present medium is $340{m}/{s}$)
A. 2068 Hz
B. 2100 Hz
C. 1903 Hz
D. 602 Hz

Answer 1

The distance between the source and observer increases with time even when they are moving with same and equal velocity and no wind is blowing. Thus, we can say that the source and observer are moving away from each other. And the velocity of sound is along the observer and opposite to the source. So, use the formula for apparent frequency which gives the relationship between original frequency and apparent frequency. Substitute the given values and find the original frequency.
Formula used:
${\nu}^{‘}= \nu(\dfrac {v-{v}_{o}}{v+{v}_{s}})$

Complete answer:
The observer and source are moving with the same speed $10 m{s}^{-1}$ and the distance between them increases with time. This means the source and observer both are moving in opposite directions and away from each other.
Given: ${\nu}^{‘}$= 1950 Hz
Velocity of source (${v}_{s}$)= Velocity of observer (${v}_{o}$)= $10 m{s}^{-1}$
Speed of sound (v)= $340{m}/{s}$
Apparent frequency is given by,
${\nu}^{‘}= \nu(\dfrac {v-{v}_{o}}{v+{v}_{s}})$
Substituting the values in above equation we get,
$1950= \nu(\dfrac {340-10}{340+10})$
$\Rightarrow 1950= \nu(\dfrac {330}{350})$
$\Rightarrow \nu= 1950(\dfrac {350}{330})$
$\Rightarrow \nu= 1950 \times 1.0606$
$\Rightarrow \nu= 2068.17 Hz$
Thus, if the apparent frequency received by an observer is 1950 Hz, then the original frequency must be 2068 Hz.

So, the correct answer is option A i.e. 2068Hz.

Note:
The above used phenomenon is called the Doppler effect. The Doppler effect is also seen in light. Any light moving towards the observer will be shifted to lower wavelengths due to Doppler effect. This is called a blue shift. Whereas, a light moving away from the observer will be shifted to longer wavelengths due to Doppler effect. This is known as a red shift. The speed of all electromagnetic waves is relatively very large as compared to the speed of source or observer, thus the apparent change in frequency is not observed easily. The apparent frequency is almost equal to the original frequency.