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Question: Two trains, each moving with a velocity of \(30m{s^{ - 1}}\) , cross each other. One of the trains g...

Two trains, each moving with a velocity of 30ms−130m{s^{ - 1}} , cross each other. One of the trains gives a whistle whose frequency is 600Hz600Hz . If the speed of sound is 330ms−1330m{s^{ - 1}} , the apparent frequency for passengers sitting in the other train before crossing would be
A. 600Hz600Hz
B. 630Hz630Hz
C. 920Hz920Hz
D. 720Hz720Hz

Explanation

Solution

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 30ms−130m{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 vs=30ms−1{v_s} = 30m{s^{ - 1}} and vo=−30ms−1{v_o} = - 30m{s^{ - 1}}
Where vs{v_s} is the velocity of source and vo{v_o} will be the velocity of observer
The apparent frequency is given by
f′=f(v−vov−vs)f' = f\left( {\dfrac{{v - {v_o}}}{{v - {v_s}}}} \right)
Putting the values of vs=30ms−1{v_s} = 30m{s^{ - 1}} and vo=−30ms−1{v_o} = - 30m{s^{ - 1}} we get
f′=600(330−(−30)330−30) =720Hz  f' = 600\left( {\dfrac{{330 - ( - 30)}}{{330 - 30}}} \right) \\\ = 720Hz \\\
Thus the apparent frequency is 720Hz720Hz

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.