A detector initially moves at constant velocity directly...

A detector initially moves at constant velocity directly toward a stationary sound source and then (after passing it) directly from it. The emitted frequency is $f$ . During the approach the detected frequency is $f_{\text { app }}^{\prime}$ and during the recession it is $f_{\text { rec. }}^{\prime}$ . If the frequencies are related by $\left(f_{\text { app }}^{\prime}-f_{\text { rec }}^{\prime}\right) / f=0.500,$ what is the ratio $v_{D} / v$ of the speed of the detector to the speed of sound?

A detector initially moves at constant velocity directly toward a stationary sound source and then (after passing it) directly from it. The emitted frequency is $f$ . During the approach the detected frequency is $f_{\text { app }}^{\prime}$ and during the recession it is $f_{\text { rec. }}^{\prime}$ . If the
frequencies are related by $\left(f_{\text { app }}^{\prime}-f_{\text { rec }}^{\prime}\right) / f=0.500,$ what is the ratio
$v_{D} / v$ of the speed of the detector to the speed of sound?

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Key Concepts

Doppler Effect

The Doppler effect refers to the change in frequency of a wave as observed by an individual moving relative to the source of the wave. For sound, this effect manifests as a higher observed frequency when the observer is moving toward the source and a lower one when moving away. This concept is fundamental in understanding how motion affects the perceived pitch of sound, and it plays a crucial role in problems involving moving detectors or sources.

Frequency Shift Equations

These equations quantify how the observed frequency changes due to the relative motion between the source and the observer. For a stationary sound source and a moving observer, the observed frequency when approaching is given by f' = f*(v + v_D)/v, and when receding by f' = f*(v - v_D)/v, where v_D is the detector's speed and v is the speed of sound. The difference between these frequencies allows one to determine the ratio of the detector's speed to the speed of sound, which is essential for solving related problems.

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