Question

A train ${{S}_{1}}$, moving with a uniform velocity of 108 kmph, approaches another train ${{S}_{2}}$ standing on a platform. An observer O moves with a uniform velocity of 36 kmph towards ${{S}_{2}}$, as shown in figure. Both the trains are blowing whistles of the same frequency 120 Hz. When O is 600m away from ${{S}_{2}}$ and distance between ${{S}_{1}},{{S}_{2}}$ is 800m, the number of beats heard by O is _________. (Speed of the sound =330 mps)

Answer 1

In the given question, we can see the observer is the same in both cases but the sources are different. The direction of travel of the two sources are also different. We need to find the frequency heard by the observer for each of the sources and then find the number of beats produced.

Formula used:
${{f}^{1}}=f(\dfrac{v+{{v}_{0}}}{v-{{v}_{s}}})$

Complete step by step answer:
Let us write down the given information first,
The speed of two sources are,
$\begin{aligned} & {{v}_{1}}=108kmph=30mps \\\ & {{v}_{2}}=0 \\\ \end{aligned}$
Next, the speed of the only observer present is 10mps.
Now, the velocity component of the source in the direction of observer will be,
${{v}_{1}}=30\cos 37=24mps$
The velocity component of the observer in the direction of source will be,
${{v}_{o}}=10\cos 53=6mps$
The frequency heard by the observer due to source 1 will be,
$\begin{aligned} & {{f}^{1}}=f(\dfrac{330+6}{330-24}) \\\ & \Rightarrow {{f}^{1}}=120(\dfrac{336}{306}) \\\ \end{aligned}$
Similarly, the frequency heard by observer due to source 2 will be,
${{f}^{1}}=f(\dfrac{330-10}{330-0})=120(\dfrac{34}{33})$
Now, beats are equal to difference between the two frequencies heard by the observer,
$\begin{aligned} & beats=120(\dfrac{336}{306}-\dfrac{34}{33}) \\\ & beats=8.128Hz \\\ \end{aligned}$
In this way, we can calculate the beats and the apparent frequencies heard by the observer when sources are at rest or at moving.

Additional information:
Doppler Effect refers to the change in wave frequency during the relative motion between a wave source and its observer. It was discovered by Christian Johann Doppler who described it as the process of increase or decrease of starlight that depends on the relative movement of the star. Doppler Effect works on both light and sound objects. For instance, when a sound object moves towards you, the frequency of the sound waves increases, leading to a higher pitch. Conversely, if it moves away from you, the frequency of the sound waves decreases and the pitch comes down. The drop in pitch of ambulance sirens as they pass by and the shift in red light are common examples of the Doppler Effect.

Note:
In the above question, when both the source and listener are moving, the velocity of the listener is added to the speed of sound because the distance between them is decreasing and the apparent frequency must be greater than the original one. If in case, they were moving away from each other, then they must be subtracted. We need to be sure about the signs of the speeds of source and observers.