Chapter 14, Problem 1CQ

(a) You are driving down the highway in your car when a police car sounding its siren overtakes you and passes you. If its frequency at rest is f₀, is the frequency you hear while the car is catching up to you higher or lower than f₀? (b) What about the frequency you hear after the car has passed you?

To determine

The frequency of the siren heard by the observer while the car sounding its siren is catching up.

Answer to Problem 1CQ

The frequency heard by you while the car is catching up is higher than the frequency $f_0$.

Explanation of Solution

The frequency of the car that is catching up can be found using the concept of Doppler effect. Doppler effect is the change in frequency heard by an observer whenever there is relative motion between a source of sound and the observer is called Doppler effect.

Formula for Doppler effect is,

$f_{Obs}=f_0\left(\frac{v+v_O}{v-v_S}\right)$

• $f_{Obs}$ is the frequency heard by you (observer).
• $f_S$ is the frequency of the Siren (source).
• $v$ is the speed of air.
• $v_{Obs}$ is the speed of your car(observer).
• $v_S$ is the speed of the police car (source).

The police car is catching you up. Relative to you, the police car is moving towards you while you are rest.

Substitute 0 for $v_O$ in the expression for Doppler effect to find $f_{Obs}$.

$\begin{array}{c}{f}_{Obs}=f_0\left(\frac{v+0}{v-v_S}\right)\\ =f_0\left(\frac{v}{v-v_S}\right)\end{array}$

The numerator contains v term and denominator contains $v-v_s$ term. The numerator is larger than the denominator. Hence, the observed frequency is greater than the frequency of the source.

Conclusion: The frequency heard by you while the car is catching up is higher than the frequency $f_0$.

To determine

The frequency of the siren heard by you while the car sounding its siren is passes away.

Answer to Problem 1CQ

The frequency heard by the you after the car passed you is lesser than the frequency $f_0$.

Explanation of Solution

The police car is move past you. Relative to you, the police car is moving away from you while you are rest.

When the source moves away from the observer, negative speed is substituted for $v_S$.

Substitute 0 for $v_O$ and $-\left(v_S\right)$ for $v_S$  in the expression for Doppler effect to find $f_{Obs}$.

$\begin{array}{c}{f}_{Obs}=f_0\left(\frac{v+0}{v-\left(-v_S\right)}\right)\\ =f_0\left(\frac{v}{v+v_S}\right)\end{array}$

The numerator contains v term and denominator contains $v+v_s$ term. The numerator is smaller than the denominator. Hence, the observed frequency is smaller than the frequency of the source.

Conclusion: The frequency heard by the you after the car passed you is lesser than the frequency $f_0$.