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Bats have poor eyesight, so they use echolocation to orient themselves and to hunt prey (insects). They emit ultrasonic sound waves as chirps and listen to hear if the reflected waves are Doppler shifted to higher or lower frequencies as they fly.

Useful Info#

The Doppler effect for sound waves tells us when a sound source and a sound observer move relative to each other at speeds $$v\_{source}$$ and $$v\_{observer}$$ respectively, the frequency observed by the observer is given by $$f\_{observer} = \frac{(1\pm \frac{v\_{observer}}{v\_{sound}})}{(1 - \mp \frac{v\_{source}}{v\_{sound}})} f_0$$ where $$f_0$$ is the frequency of emitted sound, and $$v\_{sound}$$ is the speed of sound. In the numerator of this expression a plus (minus) sign indicates that the observer is moving toward (away from) the source. In the denominator of this expression a plus (minus) sign indicates that the source is moving away from (toward) the observer. In both cases this means that the frequency increases when the objects are approaching, and decreases when they are receding from each other. (In this problem involving a reflection picked up by the original source there would actually be two Doppler shifts, with the frequency of sound reflected by the mosquito acting as the source of sound for the bat, and the bat acting as the observer of that sound.)

Part 1#

If the reflected waves from a mosquito have slightly increased in frequency: