Question

A person carrying a whistle emitting continuously a note of $272\, Hz$ is running towards a reflecting surface with a speed of $18\, km/hr$. The speed of sound in air is $345\,m/s$. The number of beats heard by him is
A. 4
B. 6
C. 8
D. 3

Answer 1

Hint: Here, the source is moving so we should consider the Doppler effect to find frequency heard by the detector. According to the doppler effect, the frequency that is heard will be different from the source frequency if there is a relative motion between the source and the detector.

Formula used:
The equation denoting the new frequency is given as
$f^{'} = \left(\dfrac{v - v_d}{v -v_s} \right)f$
Where $v$ is the speed of sound ,${v_s}$ is the speed of source, ${v_d}$ is the speed of detector and $f$ is the frequency of source sound.
Using this equation, we can find the frequency that is heard after reflection. By subtracting the source frequency from the frequency heard we can get the number of beats.

Complete step by step answer:
The frequency of the whistle is given as $272\,Hz$.
This is the source frequency
$f = 272\,Hz$
Velocity of the source.
${v_s} = 18\,km/hr$
In order to convert $km/hr$ into $m/s$ we need to multiply by $\dfrac{{1000}}{{3600}}$
Since $1\,km = 1000\,m$
and$1\,hr = 3600\,s$
Speed of sound in air is given as
$v = 345\,m/s$
Here, since the source is moving we should consider the doppler effect to find frequency heard by the detector.
According to the doppler effect, the frequency that is heard will be different from the source frequency if there is a relative motion between the source and the detector.
The equation denoting the new frequency is given as
$f^{'} = \left(\dfrac{v - v_d}{v - v_s} \right)f$
Where $v$ is the speed of sound, ${v_s}$ is the speed of source, ${v_d}$ is the speed of detector and $f$ is the frequency of source sound.
Here, since sound is reflected, we can take the value of ${v_d}$ the same as the value of ${v_s}$.
But the direction of reflected sound is opposite to the direction of the source sound. Thus, we need to
take negative values of speed of source as the speed of the detector.
That is, ${v_d} = - {v_s}$
Thus, our equation can be written as
$f^{'} = \left(\dfrac{v + v_s}{v - v_s} \right)f$
Substitute the given values, then we get
$f' = \left(\dfrac{345 + 5}{345 - 5} \right) \times 272 = 280\,Hz$
The number of beats per second is equal to the difference in the frequencies of the two ways.

Therefore, Number of beats = $f' - f = 280 - 272 = 8$. So, the correct answer is option C.

Note:
While substituting for the value of the velocity of receiving sound always take care of the sign. The direction of propagation of the source waves is taken as a positive direction. The magnitude of the speed of reflected sound will be the same as that of the source speed due to reflection but the direction is reversed so we need to take the negative sign.