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TFM

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**[SOLVED] Horsehoe Bats, Insects, Speed... and the Doppler Effect**

## Homework Statement

Horseshoe bats (genus

*Rhinolophus*) emit sounds from their nostrils, 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 [tex] v_b_a_t[/tex] emits sound of frequency [tex] f_b_a_t [/tex] ; the sound it hears reflected from an insect flying toward it has a higher frequency [tex] f_r_e_f_c [/tex] .

If the bat emits a sound at a frequency of 80.8 kHz and hears it reflected at a frequency of 83.1 kHz while traveling at a speed of 4.2 m/s, calculate the speed of the insect.

Use 344 m/s for the speed of sound in air. Express your answer using two significant figures.

## Homework Equations

Doppler Shift Equations:

[tex] f_I = (\frac{v + v_I}{v + v_b_a_t})f_b_a_t [/tex]

[tex] f_r_e_f = (\frac{v + v_b_a_t}{v + v_I})f_I [/tex]

## The Attempt at a Solution

the answer I got by using these two equations was 10.45, got via:

**firstly, I inserted the values into the top equation, to get [tex] f_I [/tex] in terms of [tex] v_I [/tex] :**

[tex] f_I = (\frac{v + v_I}{v + v_b_a_t})f_b_a_t [/tex]

[tex] f_I = (\frac{344 + v_I}{344 + (-4.2)})80.8 [/tex]

[tex] f_I = (\frac{344 + v_I}{339.8})80.8 [/tex]

[tex] f_I = (\frac{344}{339.8} + \frac{v_I}{339.8})80.8 [/tex]

[tex] f_I = (81.7987 + \frac{80.8v_I}{339.8}) [/tex]

[tex] f_I = (81.7987 + 0.2672v_I) [/tex]

**I then rearranged the second equation:**

[tex] f_r_e_f = (\frac{v + v_b_a_t}{v + v_I})f_I [/tex]

[tex] 83.1 = (\frac{344 + 4.2}{344 + v_I})f_I [/tex]

[tex] 83.1 = (\frac{348.2}{344 + v_I})f_I [/tex]

[tex] 83.1*(344 + v_I) = 348.2f_I [/tex]

[tex] 28586.4 + 83.1v_I) = 348.2f_I [/tex]

**Now inserting [tex] f_I [/tex]:**

[tex] 28586.4 + 83.1v_I) = 348.2((81.7987 + 0.2672v_I)) [/tex]

[tex] 28586.4 + 83.1v_I) = ((28482.31 + 93.045v_I)) [/tex]

[tex] 104.0909 = 9.94v_I [/tex]

Giving the Insects speed as 10.46

Any ideas where I have gone wrong?

TFM

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