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A bat flies toward a moth at speed 8.0 m/s while the moth is flying toward the bat at speed 5.0 m/s. The bat emits a sound wave of 51.35 kHz. What is the frequency of the wave detected by the bat after it reflects off the moth?

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This is the mother of all doppler problems. First, the sound is source shifted by the bat's 8.0 m/s, then observer and source shifted as it reflects from the moth's 5.0 m/s, and finally the bat is a moving observer to the reflected waves.:

1. Moving source from bat:

f' = f/(1 - vs/v) = (51.35 kHz)/(1-(8.0 m/s)/(343 m/s) = 52.57627 kHz

2. Moving observer from moth:

f' = f ( 1 + v0/v) = (52.57627 kHz)(1+(5.0 m/s)/(343 m/s)) = 53.3427 kHz

3. Moving source from moth:

f' = f/(1 - vs/v) = (53.3427 kHz)/(1-(5.0 m/s)/(343 m/s)) = 54.13178 kHz

And Finally,

4. Moving observer from bat:

f' = f ( 1 + v0/v) = (54.13178 kHz)(1+(8.0 m/s)/(343 m/s)) = 55.3943 kHz = 55.39 KHz

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