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A stationary police car emits a sound of frequency 1200 Hz that bounces off a car on the highway and returns with a frequency of 1250 Hz. The police car is right next to the highway, so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving toward or away from the police car? (b) What frequency would the police car have received if it had been traveling toward the other car at 20.0 m/s?

Short Answer

Expert verified

(a) Speed of car is 7.02m/s and it is moving towards the police officer.

(b) Frequency received by the police car is 1404Hz

Step by step solution

01

Step 1:

Given data:

As by the doppler effect:

fL=v+vLv+vSfS

fL=Frequency observed by the listener

v=speed of sound

vL=speed of listener

fS=frequency of source

vS=speed of the source of sound

And here vL is positive as the velocity of the listener from Listener to Source and vS is positive as the velocity of the source from Listener to Source and velocity is negative.

02

Step 2:

(a) Consider the car as if it is moving away, and if the car has a negative speed, it is moving in the opposite direction.

vcarvelocity of moving car

Thus, two doppler shifts:

First case: vLit is negative as the listener is moving away from the source.

fL=v+vLv+vSfS=v-vCarv+01200Hz

Second case: source is the moving car and the police are the listener

fsfirst shifted frequency, vL0,vSit is positive as the source is moving away from the listener.

role="math" localid="1668071173728" fL=v+vLv+vSfS=v+0v+vCar×vvCarv+0(1200Hz)=1250HzvvvCarvv+vCar=1250Hz1200Hz=2524,solveforvCar25v+25vCar=24v24vCar49vCar=-vSovCar=7.02m/s

Hence, the speed of the car is 7.02m/s and it is moving toward the police officer.

03

Step 3:

(b) As vcarvelocity of the moving car

Thus, two doppler shifts:

First case: vLit is positive as the listener is moving towards from the source,vs=vpoliceit is negative as moving towards the listener.

fL=v+vLv+vSfS=v+7.02m/sv20m/s(1200Hz)=1300.07Hz

Second case: source is the moving car and police is the listener

fS=1300.07Hzfirst shifted frequency, vL=20m/sit is positive as it is moving towards the source, vSit is negative as the source is moving towards the listener.

fL=v+vLv+vSfS=v+vpolicevvCar(1300Hz)=344m/s+20m/s344m/s7.02(1300Hz)=1404Hz

Hence, the frequency received by the police car is 1404Hz.

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