Question

A police car is moving at 30m/s, chases a motorcyclist. The policeman sounds his horn at 180Hz while both of them move towards a stationary siren of frequency 160Hz. Calculate the speed of the motorcyclist, if it is given that he does not observe any beats (take, the speed of sound=330m/s).
A 25 m/s
B 30 m/s
C 40 m/s
D 35 m/s

Answer 1

The modulus of the difference between the two variable frequencies of the wave gives us the beat frequency, $f_B$ =|$f_1$ - $f_2$ |. Here $f_1$ and $f_2$ are the frequency of two waves. To calculate frequency we use the formula f=${f_0}\left( {\dfrac{{u \pm {v_O}}}{{u \pm {v_S}}}} \right)$.
Here f is the frequency, $f_0$ is the frequency of source, $v_s$ is the velocity of the source, $v_o$ is the velocity of the observer and u is the speed of surrounding air.
If the velocity of the observer is towards the source then the numerator contains a positive sign and vice versa. If the velocity of the source is away from the observer the denominator contains a positive sign and vice versa.

Complete step by step answer:
We use the formula f=${f_0}\left( {\dfrac{{u \pm {v_O}}}{{u \pm {v_S}}}} \right)$. ---- (1)
Here f is the frequency, f0 is the frequency of source, vS is the velocity of the source, vO is the velocity of the observer and u is the speed of surrounding air.
Case 1: For a police car, f=f1, f0=180, vO=v, u=330, vS=30 towards the motorcyclist.
Substituting values in equation (1),
${f_1} = 180\left( {\dfrac{{330 - v}}{{330 - 30}}} \right) \\\ {f_1} = 180\left( {\dfrac{{330 - v}}{{300}}} \right) \\\$
Case 2: For siren, f=f2, f0=160, vO=u, u=330, vS=0 .
Substituting values in equation (1),
${f_2} = 160\left( {\dfrac{{330 + v}}{{330 + 0}}} \right) \\\ {f_2} = 160\left( {\dfrac{{330 + v}}{{330}}} \right) \\\$
Since there is no beat, f1= f2
$180\left( {\dfrac{{330 - v}}{{300}}} \right) = 160\left( {\dfrac{{330 - v}}{{330}}} \right) \\\ \implies 9\left( {\dfrac{{330 - v}}{{10}}} \right) = 8\left( {\dfrac{{330 - v}}{{11}}} \right) \\\ \implies 32670 - 99v = 26400 + 80v \\\ \implies 179v = 6270 \\\ \implies v = \left( {\dfrac{{6270}}{{179}}} \right) \\\$
$\therefore v =35.02 m/s$
Therefore velocity of the motorcyclist is approx. 35 m/s.

So, the correct answer is “Option D”.

Additional Information:
When two waves of nearly equal frequencies travelling in a medium along the same direction meet at a point beat is produced. When two sound waves of dissimilar frequency approach our ears the alternative destructive and constructive interference causes sound to alternatively be loud and soft this phenomenon is known as beating.

Note:
The sign for velocity of observer and source should be kept into consideration to avoid calculation mistakes. Frequency is a positive quantity and therefore beat it is the modulus of difference of the frequencies of waves.