91影视

The frequency emitted from the bat ${\mathrm{f}}_{\mathrm{bat}}$, and the frequency detected by the bat after being reflected from the insect is f_refl ,In the first case, the bat will be the source of sound that moves towards the listener "the insect"

The formula that describes the frequency detected by the insect, is ${\mathrm{f}}_1={\mathrm{f}}_{\mathrm{s}}\left(\frac{\mathrm{v}卤{\mathrm{v}}_{\mathrm{L}}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{s}}}\right)$ where, v, the speed of sound in the air, VL = v_{in sect} the speed of the insect, vs = ${\mathbf{v}}_{\mathbf{bat}}$, the speed of the source "the bat", the frequency detected by the insect, fs = f_bat , the frequency emitted from the bat.

Therefore, equation can be written as role="math" localid="1668143929396" ${\mathrm{f}}_{\mathrm{L}}={\mathrm{f}}_{\mathrm{bat}}\left(\frac{\mathrm{v}卤{\mathrm{v}}_{\mathrm{insect}}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}\right)$(1)

Now, after the sound reflects from the insect, it will travel towards the bat and the frequency detected by the bat in so the equation (1) can be written as

${\mathrm{f}}_{\mathrm{refl}}={\mathrm{f}}_{\mathrm{L}}\left(\frac{\mathrm{v}+{{\mathrm{v}}_{\mathrm{bat}}}_{}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{insect}}}\right)$(2)

Substitute ${\mathrm{f}}_{\mathrm{L}}$in equation 2 from 1

${\mathrm{f}}_{\mathrm{refl}}={\mathrm{f}}_{\mathrm{bat}}\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{insect}}}{\mathrm{v}-{\mathrm{v}}_{\mathrm{bat}}}\right)\left(\frac{\mathrm{v}+{{\mathrm{v}}_{\mathrm{bat}}}_{}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{insect}}}\right)$

$$\begin{gathered} \frac{{\mathrm{f}}_{\mathrm{refl}}}{{\mathrm{f}}_{\mathrm{bat}}}=\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{insect}}}{\mathrm{v}鈭�{\mathrm{v}}_{{}_{\mathrm{insect}}}}\right)\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}\right) \\ \frac{{\mathrm{f}}_{\mathrm{refl}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)=\left(\frac{\mathrm{v}+{\mathrm{v}}_{{\mathrm{insect}}_{}}}{\mathrm{v}鈭�{\mathrm{v}}_{{\mathrm{insect}}_{}}}\right) \end{gathered}$$

Let$\frac{{\mathrm{f}}_{\mathrm{refl}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)$ be x hence.

$$\begin{gathered} \mathrm{x}=\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{insect}}}{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{insect}}}\right) \\ \mathrm{xv}鈭�{\mathrm{v}}_{\mathrm{insect}}\mathrm{x}=\mathrm{v}+{\mathrm{v}}_{\mathrm{insect}} \\ {\mathrm{v}}_{\mathrm{insect}}+{\mathrm{v}}_{\mathrm{insect}}\mathrm{x}=\mathrm{vx}鈭�\mathrm{v} \\ {\mathrm{v}}_{\mathrm{insect}}(1+\mathrm{x})=\mathrm{v}(\mathrm{x}鈭�1) \\ {\mathrm{v}}_{\mathrm{insect}}=\frac{\mathrm{v}(\mathrm{x}鈭�1)}{(1+\mathrm{x})} \end{gathered}$$

Now let鈥檚 replace x with$\frac{{\mathrm{f}}_{\mathrm{refl}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)$

${\mathrm{v}}_{\mathrm{insect}}=\frac{\mathrm{v}\left(\frac{{\mathrm{f}}_{\mathrm{reff}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)鈭�1\right)}{\frac{{\mathrm{f}}_{\mathrm{ref}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)+1}$

Note that $1=\frac{{\mathrm{f}}_{\mathrm{bat}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)$Hence,

${\mathrm{v}}_{\mathrm{insect}}=\frac{\mathrm{v}\left(\frac{{\mathrm{f}}_{\mathrm{ret}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)鈭�\frac{{\mathrm{f}}_{\mathrm{bat}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)\right)}{\frac{{\mathrm{f}}_{\mathrm{ref}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)+\frac{{\mathrm{f}}_{\mathrm{bat}}}{{\mathrm{f}}_{\mathrm{bat}}}\left(\frac{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}}\right)}$

${\mathrm{v}}_{\mathrm{insect}}=\frac{\mathrm{v}\left({\mathrm{f}}_{\mathrm{ref}}\left(\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}\right)鈭�{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)\right)}{{\mathrm{f}}_{\mathrm{refl}}\left(\mathrm{v}鈭�{\mathrm{v}}_{\mathrm{bat}}\right)+{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)}$(1)

$$\begin{gathered} {\mathrm{v}}_{\mathrm{in}\mathrm{sect}}=344脳\frac{\left[\left(83.5脳{10}^3\right)\left(344\right)-\left(3.9\right)\right]-\left[\left(80.7脳{10}^3\right)\left(344\right)+\left(3.9\right)\right]}{\left[\left(83.5脳{10}^3\right)\left(344\right)-\left(3.9\right)\right]+\left[\left(80.7脳{10}^3\right)\left(344\right)+\left(3.9\right)\right]} \\ =1.97\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, 1.97m/s is the speed of the insect.