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Question

A police car moving at 22 m/s, chase a motorcyclist. The police man sounds his horn at 176 Hz, while both of them move towards a stationary siren of frequency 165 Hz. Calculate the speed of the motorcyclist, if he does not observe any beats: (velocity of sound in air = 330 m/s)
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A
33 m/s
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B
22 m/s
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C
zero
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D
11 m/s
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Solution

The correct option is C 22 m/s
In first situation, a police car (source of sound) with a speed vs, is approaching a motorcycle ( observer), which is moving away from police car with a speed of vm.
Therefore apparent frequency of sound heard by motorcyclist,
n=ncar(vvmvvs) ...................eq1 .
In second situation, motorcyclist (observer) is approaching a stationary siren (source), with a speed of vm.
Therefore apparent frequency of sound heard by motorcyclist,
n′′=nsiren(v+vmv) ...................eq2 ,.
Now as no beats are observed by motorcyclist, this is possible only when difference in frequencies heard by motorcyclist is zero,
i.e. nn′′=0,
or n=n′′ ,
or ncar(vvmvvs)=nsiren(v+vmv)
Given n=176Hz, n′′=165Hz, v=330m/s, vs=22m/s,ncar=176Hz and nsiren=165Hz
Hence, 176(vvm33022)=165(v+vm330) ,
or (vvmv+vm)=165176×308330=7/8 ,
or 15vm=v,
Speed of sound in air v=330 m/s
or vm=330/15=22m/s

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