A police car moving at `22m//s`, chases a motorcyclist, the police man sounds his horn at `176 Hz`, while both of them move towards a stationary siren of frequency `165 H`z. Calculate the speed of the motorcycle, if it is given that he does not observes any beat

Options

A. `11 m-s^(-1)`
B. `22 m-s^(-1)`
C. `33 m-s^(-1)`
D. `44 m-s^(-1)`

Answer

B

Explanation

Given : frequnecy of police horn `(v_(h)) - 176 Hz`
Velocity of police car `(v_(P)) = 22 m -s^(-1)`
Velocity of motorcycle `= v_(m)`
Frequency of stationary siren `(v_(s)) = 165 Hz`
We know from the Dopplar's effect that apparent frequnecy heard by the motorcyclist from the police car, when both are moving in the same direction.
`(v_(1))=((v-v_(m))/(v-v_(P)))xxv_(h)=(330-v_(m))/(330-22)xx176=(330-v_(m))/(308)xx176`
Similarly, apparent frequency heard by motorcycle from the stationary siren, when he is moving towards it
`(v_(2)) = ((v+v_(m))/(v-v_(s)))xxv_(s)=((330+v_(m))/(330-0))xx165=(330+v_(m))/(330)xx165`
Since the motorcycle does not observe any beat,
therefore, `v_(1) - v_(2) = 0` or `v_(1) - v_(2)`
or `(330-v_(m))/(308) xx176=((330+v_(m))/(330))xx165`
or `(300-v_(m))/(330+v_(m))=(165)/(176) xx (308)/(330) = 7/8`
or `2640-8v_(m) = 2310+7v_(m)`
or `15v_(m) = 2640-2310 = 330`
or `v_(m) = (330)/(15) = 22 m-s^(-1)`