A police car siren emits a sound at 600 Hz.If an observer moves toward the source at 50 km/hr and hears a siren frequency of 678 Hz,calculate the speed of the police car and whether it is approaching or receding from the observer (assume the speed of sound in air = 343 m/s).
A)99.0 km/hr receding
B)144 km/hr approaching
C)99.0 km/hr approaching
D)144 km/hr receding
E)not possible for any reasonable police car speed

Question

Quizplus · Accepted Answer

99.0 km/hr approachingTo solve this problem, we use the Doppler effect formula for sound. The observed frequency $f'$ is given by $f' = f \frac{v + v_o}{v - v_s}$, where $f$ is the source frequency, $v$ is the speed of sound, $v_o$ is the observer's speed towards the source, and $v_s$ is the source's speed towards the observer. Given that $f = 600$ Hz, $f' = 678$ Hz, $v = 343$ m/s, and $v_o = 50$ km/hr (which is $13.89$ m/s), we can rearrange the formula to solve for $v_s$.Rearranging the formula gives us $v_s = v\left(\frac{f}{f'} - 1\right) - v_o\left(\frac{f}{f'}\right)$. Plugging in the given values, we find that $v_s$ is positive, indicating that the source (police car) is moving towards the observer. Converting $v_s$ from m/s to km/hr gives us approximately $99.0$ km/hr, indicating the police car is approaching the observer.

A Police Car Siren Emits a Sound at 600 Hz