Q69P - University Physics with Modern Physics

Question

BIO Horseshoe bats (genus Rhinolophus) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey’s speed. (The “horseshoe” that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A Rhinolophus flying at speed v_bat  emits sound of frequency f_bat ; the sound it hears reflected from an insect flying toward it has a higher frequency f_refl .(a) Show that the speed of the insect is

${\mathrm{v}}_{\mathrm{in} \mathrm{sect}}=\frac{\mathrm{v}\left({\mathrm{f}}_{\mathrm{refl}}\left(\mathrm{v}-{\mathrm{v}}_{\mathrm{bat}}\right)-{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)\right)}{{\mathrm{f}}_{\mathrm{refl}}\left(\mathrm{v}-{\mathrm{v}}_{\mathrm{bat}}\right)+{\mathrm{f}}_{\mathrm{bat}}\left(\mathrm{v}+{\mathrm{v}}_{\mathrm{bat}}\right)}$$cv$

where v is the speed of sound. (b) If f_bat = 80.7 kHz,  f_refl =83.5 kHz, and  v_bat = 3.9 m>s, calculate the speed of the insect.

Step-by-Step Solution

Answer

A) We used the Doppler effect to find the frequency detected by the insect, then we used that frequency as the frequency of the sound reflected from the insect toward the bat, after that, we used some algebraic manipulation to get the result we wanted. 

b) ${\mathrm{v}}_{\mathrm{in} \mathrm{sect}}=1.97\mathrm{m}/\mathrm{s}$ is the speed of the insect.

STEP 1 Formula of the frequency detected by the insect

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 ${\mathrm{f}}_1={\mathrm{f}}_{\mathrm{s}}\left(\frac{\mathrm{v}\pm {\mathrm{v}}_{\mathrm{L}}}{\mathrm{v}\mp {\mathrm{v}}_{\mathrm{s}}}\right)$ 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 ${\mathrm{f}}_{\mathrm{L}}={\mathrm{f}}_{\mathrm{bat}}\left(\frac{\mathrm{v}\pm {\mathrm{v}}_{\mathrm{insect} }}{\mathrm{v}\mp {\mathrm{v}}_{\mathrm{bat}}}\right)$ (1)

STEP 2 Calculate the speed of the insect

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 

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

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

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

Step 3 Equation to solve for

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

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

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

Now let’s replace x with $\frac{{\mathrm{f}}_{\mathrm{refl} }}{{\mathrm{f}}_{\mathrm{bat} }}\left(\frac{\mathrm{v}-{\mathrm{v}}_{\mathrm{bat} }}{\mathrm{v}+{\mathrm{v}}_{\mathrm{bat} }}\right)$

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

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,

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

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

Step 4 Substitute the values in equation 1

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

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