A particular bat emits ultrasonic waves with a frequency of 56.00 kHz. The bat is traveling at 20.00 m/s toward a moth, which is flying away from the bat at 8.00 m/s. The speed of sound is 340.0 m/s.

(a) Assuming the moth could detect the waves, what frequency waves would it observe?

(b) The waves reflect off the moth and are detected by the bat. What frequency are the waves detected by the bat?

Part (a). We use the general Doppler equation:

f^/ = f (v +/- v_O) / (v -/+ v_s )

where f = 56.0 kHz and v = 340 m/s
v_O = 8.00 m/s (use the minus sign - moving away)
v_s = 20.0 m/s (use the minus sign - moving toward)

Combining this gives:

f^/ = 56.0 kHz (340 - 8.00) / (340 - 20.0)

f^/ = 58.1 kHz

Part (b). Once again we use the general Doppler equation, but this time the bat is the observer and the moth acts as the source.

f^// = f^/ (v +/- v_O) / (v -/+ v_s )

where f^/ = 58.1 kHz and v = 340 m/s
v_O = 20.0 m/s (use the plus sign - moving toward)
v_s = 8.00 m/s (use the plus sign - moving away)

These give:

f^// = 58.1 kHz (340 + 20.0) / (340 + 8.00)

f^// = 60.1 kHz

The bat detects a 60.1 kHz wave, which it could use to figure out how fast the moth is flying.