Question

You are driving along a highway at 35.0 m/s when you hear the siren of a police car approaching you from behind at constant speed and you perceive the frequency as 1340 Hz. You are relieved that he is in pursuit of a different driver when he continues past you, but now you perceive the frequency as 1300 Hz.What is the speed of the police car?

Answer 1

Answer:

40.13491 m/s

Explanation:

$v_r$ =  My speed = 35 m/s

v = Speed of sound in air = 343 Hz

$v_s$ = Speed of the police car

When the car is approaching

$f=f'\dfrac{v-v_r}{v-v_s}\\\Rightarrow 1340=f'\dfrac{343-35}{343-v_s}$

When the car is receding

$f=f'\dfrac{v+v_r}{v+v_s}\\\Rightarrow 1300=f'\dfrac{343+35}{343+v_s}$

Dividing the equations

$\dfrac{1340}{1300}=\dfrac{f'\dfrac{343-35}{343-v_s}}{f'\dfrac{343+35}{343+v_s}}\\\Rightarrow \dfrac{1340}{1300}=\dfrac{22\left(v_s+343\right)}{27\left(-v_s+343\right)}\\\Rightarrow -36180v_s+12409740-12409740=28600v_s+9809800-12409740\\\Rightarrow \frac{-64780v_s}{-64780}=\frac{-2599940}{-64780}\\\Rightarrow v_s=\frac{129997}{3239}\\\Rightarrow v_s=40.13491\ m/s$

The speed of the police car is 40.13491 m/s