Question

Two trains move towards each other with the same speed. The speed of sound is 340 m/sec. If the height of the tone of the whistle of one of them heard on the other changes 9/8 times, then the speed of each train should be
A. 2000 m/sec
B. 200 m/sec
C. 20 m/sec
D. 2 m/sec

Answer 1

Hint: As the two trains are moving towards each other, so there is relative motion between the trains. We use the Doppler’s effect to observe the change in the frequency of the sound coming out of the source.

Formula used:
\[{f_{ap}} = {f_o}\left( {\dfrac{{v \pm {v_o}}}{{v \pm {v_s}}}} \right)\]
where \[{f_{ap}}\] is the apparent frequency heard by the listener moving with speed \[{v_o}\]with respect to the source which is moving with speed \[{v_s}\], \[{f_o}\] is the original frequency and $v$ is the speed of sound in air.

Complete step by step solution:
It is given that the trains are moving towards each other. So, the source and the observer are in relative motion. When the source and the observer is moving towards each other then the Doppler’s effect formula becomes,
\[{f_{ap}} = {f_o}\left( {\dfrac{{v + {v_o}}}{{v - {v_s}}}} \right)\]
Here, the speed of the source and the observer is the same. Let the speed of trains be equal to v’.

It is given that the apparent frequency is \[\dfrac{9}{8}\] times the original frequency.
\[{f_{ap}} = \dfrac{9}{8}{f_o}\]
Putting in the Doppler’s equation formula, we get
\[\dfrac{9}{8}{f_o} = {f_o}\left( {\dfrac{{v + v'}}{{v - v'}}} \right) \\ \]
\[\Rightarrow 9v - 9v' = 8v + 8v' \\ \]
\[\Rightarrow v = 17v' \\ \]
\[\Rightarrow v' = \dfrac{1}{{17}}v\]
The speed of the sound in air is given as 340 m/s.
So, the speed of the trains will be,
\[v' = \dfrac{1}{{17}} \times 340\,m/s \\ \]
\[\therefore v' = 20\,m/s\]
Hence, the trains are moving towards each other with speed 20 m/s.

Therefore, the correct option is C.

Note: The trains are in relative motion, but we should not use the relative velocity by assuming one of them is at rest and the other is moving with respect to the train taken as reference.