Question

The whistles of the two trains, $A$ and $B$ respectively, each have a frequency of $392\text{Hz}$. $A$ is stationary and $B$ is moving towards the right (away from $A$) at a speed of $35.0\text{m/s}$. A listener is between the two whistles and is moving towards the right with a speed of $15.0\text{m/s}$. No wind is blowing. How many number of beats are heard by the listener in $10$ seconds ?

Answer 1

Doppler effect equation :-
$f_{abs.}=f_{source}\left(\frac{V_{sound}\pm V_{obs}}{V_{sound}\mp V_{source}}\right)$
Frequency of whistle A as heard by listener :-
$f_A=\left(\frac{343-15}{343+0}\right)\times 392=374.857\text{Hz}$
Frequency of whistle $B$ as heard by the
listener:-
$f_B=\left(\frac{343+15}{343+35}\right)392=\frac{358}{378}\times 392$
$f_B=371.259$
Beat frequency detected by the listener,
$f_{beat}=|f_B-f_A|$
$=|371.259-374.857|$
$=3.597\text{Hz}\approx 3.6\text{Hz}$
Number of beats heard in 10 seconds is $36$