Question

A train moves towards a stationary observer with a speed of 34m/s. The train sounds a whistle and its frequency registered by the observer is f1. If the train’s speed is reduced to 17m/s, then the frequency registered is f2. If the speed of the sound is 340m/s, then the ratio f1/f2 is:
A. 18/19
B. 1/2
C. 2
D. 19/18

Answer 1

Hint: Use the doppler shift formula:
When the source is moving towards the stationary source-\[{f_{obs}} = {f_o}\left[ {\dfrac{v}{{v - {v_s}}}} \right]\]
When the source is moving away from stationary source \[{f_{obs}} = {f_o}\left[ {\dfrac{v}{{v + {v_s}}}} \right]\]
Where v is the speed of sound, vs is the speed of source.

Complete step by step answer:When a source of sound or observer listening to sound is moving away or towards one another there is a shift in frequency from its original value. Therefore, the observer hears a different sound then the original sound. This is known as Doppler Effect. And the change in frequency is known as Doppler Shift. It is one of the important concepts in the field of Astronomy and explains a quite number of observed effects in the universe.
Doppler derived the relation between observed and original frequency for various conditions viz. Stationary source and observer moving towards or from the observer, Stationary Observer and source is from or towards the source, Observer and source both moving away from each other, observer and source both moving towards each other.
Here the source is moving towards stationary observer, so,
     \[{f_{obs}} = {f_o}\left[ {\dfrac{v}{{v - {v_s}}}} \right]\]
First the train (source) is moving with v1=34m/s then train is moving with v2=17m/s, therefore,
     \[
  {f_1} = {f_o}\left[ {\dfrac{v}{{v - {v_1}}}} \right] \\
  {f_2} = {f_o}\left[ {\dfrac{v}{{v - {v_2}}}} \right] \\
  \dfrac{{{f_2}}}{{{f_1}}} = \dfrac{{v - {v_1}}}{{v - {v_2}}} \\
 \]
Substituting the data, we get,
     \[
  \dfrac{{{f_2}}}{{{f_1}}} = \dfrac{{340 - 34}}{{340 - 17}} = \dfrac{{306}}{{323}} \\
  \dfrac{{{f_2}}}{{{f_1}}} = \dfrac{{18}}{{19}} \\
 \]
Hence, the correct answer is option A.

Note:Many times students get confused whether there was a negative or positive sign in the denominator. To overcome this, think practically. When a source is moving towards the observer, the original sound wave will now have less distance to fit in after time t (tending to zero) as source would have moved forward distance x (tending to zero). So obviously the frequency will increase and wavelength will decrease. Hence, put the signs accordingly. Same explanation can be given for other cases mentioned in the step by step solution.