Question

A bat locates insects by emitting ultrasonic “chirps” and then listening for echoes from the bugs. Suppose a bat chirp has a frequency of 25 kHz. How fast would the bat have to fly, and in what direction, for you to just barely be able to hear the chirp at 20 kHz?

Short answer

The bat should fly at the speed of$86m/s$ away from you.

Step-by-step solution

Given information

The frequency of bat chirps is $f_0=25kHz=25000Hz$

A bat must fly away from you so that the chirp frequency observed by you is less than 25 kHz, that is, $f_{-}=20kHz=20000Hz$.

Speed of sound in air is$v=343m/s$.

Speed of bat

Doppler effect for receding source is given by$f_{-}=\frac{f_0}{1+{\displaystyle \frac{v_s}{v}}}$.

Substitute the given values

$$\begin{gathered} 20000=\frac{25000}{1+{\displaystyle \frac{vs}{343}}} \\ \frac{v_s}{343}=1.25-1 \\ {v}_s=0.25\times 343 \\ \approx 86m/s \end{gathered}$$

Therefore, the bat should fly at the speed of $86m/s$away from you.