Question

Two trains are moving towards each other at speeds of $20m{s^{ - 1}},15m{s^{ - 1}}$ relative to the ground. The first train sounds a whistle of frequency $600Hz$ the frequency of the whistle heard by a passenger in the second train before the train meet is (speed of sound in air is $340m{s^{ - 1}}$ )
(A) $600Hz$
(B) $85Hz$
(C) $645Hz$
(D) $666Hz$

Answer 1

In order to solve this question, we should know that when there is relative motion between source of sound and the observer the apparent frequency to the observer is different from the actual frequency of the sound and this phenomenon is known as Doppler effect of sound. Here, we will use the general formula of apparent frequency of sound in the Doppler effect to solve this question.

Formula Used:
The apparent frequency in Doppler effect is calculated by using the formula
${f_0} = \dfrac{{v + {v_0}}}{{v - {v_s}}}{f_s}$
where,
${f_0}$ is the apparent frequency heard by the observer.
${f_s}$ is the actual frequency of sound.
$v$ is the velocity of sound in air.
${v_0}$ is the velocity of the observer with respect to the ground frame.
${v_s}$ is the velocity of the source of sound with respect to ground frame.

Complete step by step solution:
According to the question we have given that,
${f_s} = 600Hz$ actual frequency of sound.
$v = 340m{s^{ - 1}}$ speed of sound in air.
${v_0} = 15m{s^{ - 1}}$ is the velocity of the observer with respect to the ground frame.
${v_s} = 20m{s^{ - 1}}$ is the velocity of the source of sound with respect to the ground frame.
Let ${f_0}$ be the frequency heard by the observer so using formula,
${f_0} = \dfrac{{v + {v_0}}}{{v - {v_s}}}{f_s}$ putting the values of all parameters we get,
${f_0} = \dfrac{{340 + 15}}{{340 - 20}}(600)Hz$
${f_0} = 600 \times 1.109$
$\Rightarrow {f_0} = 665.625Hz$
or we can say,
${f_0} \approx 666Hz$
Hence, the correct option is (D) $666Hz$

Note: It should be remembered that, here both the source of sound and the observer were moving towards each other, if they were moving apart from each other the positive and negative signs in the formula will be changed to negative and positive respectively. and when a source of sound moves towards the observer, the apparent frequency is always larger than the original frequency of sound.