An acoustical motion detector emits a 50 kHz signal and receives the echo signal. If the echoes have Doppler shift frequency components departing from 50 kHz by more than 100 Hz, a "moving object" is registered. For a sound velocity in air of 330 m/sec, calculate the speed with which an object must move toward (or away from) the detector in order to be registered as a "moving object".
Text Solution
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Consider a source emitting sound of frequency v. The Doppler effect has it that if an observer moves with velocity v toward the source he will detect the frequency as
c being the speed of sound propagation. On the other hand, if the source moves with velocity v toward the observer, who is stationary, then
Thus the object, moving toward the detector, receives a signal of frequency
and the signal after reflection by the object is detected by the detector as having frequency
For the moving object to be registered, we must have 
, where 
. Then
Or
as 
. Hence the object must be moving toward or receding from the detector at
for it to be registered.
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