Question

Two trains, each moving with a velocity of $30m{s^{ - 1}}$ , cross each other. One of the trains gives a whistle whose frequency is $600Hz$ . If the speed of sound is $330m{s^{ - 1}}$ , the apparent frequency for passengers sitting in the other train before crossing would be
A. $600Hz$
B. $630Hz$
C. $920Hz$
D. $720Hz$

Answer 1

Hint: In this question, we have been asked to evaluate the apparent frequency for passengers sitting in the other train before crossing such that two trains are moving in opposite direction with the velocity of $30m{s^{ - 1}}$ and blowing their whistle. In this case, we need to use the formula for apparent frequency and use the velocities mentioned with appropriate signs.

Complete step by step answer:
Since the source and observer are moving towards each other, the sign of the velocity of the source will be taken as positive and that of the observer will be negative. To determine this we define an axis from source to observer. Now any particle that goes in the same direction if the axis that the direction of source to observe will have positive velocity and the particle going in the opposite direction will have negative velocity.
Thus ${v_s} = 30m{s^{ - 1}}$ and ${v_o} = - 30m{s^{ - 1}}$
Where ${v_s}$ is the velocity of source and ${v_o}$ will be the velocity of observer
The apparent frequency is given by
\[f' = f\left( {\dfrac{{v - {v_o}}}{{v - {v_s}}}} \right)\]
Putting the values of ${v_s} = 30m{s^{ - 1}}$ and ${v_o} = - 30m{s^{ - 1}}$ we get
$
  f' = 600\left( {\dfrac{{330 - ( - 30)}}{{330 - 30}}} \right) \\
   = 720Hz \\
 $
Thus the apparent frequency is $720Hz$

So, the correct answer is “Option D”.

Note:
The frequency heard by the listener will not be the same as that being emitted. The concept of determining the signs in the formula for apparent frequency should also be kept in mind as this is useful for solving these types of problems. Here an apparent frequency will be heard. Students should know when to use positive signs and when to use negative signs in the formula for apparent frequency.