Speed of moth related to Doppler Effect

[songoku]

Homework Statement

Sound wave produced by a bat has a frequency of 222 kHz and wavelength of $1.5 x 10^{-3}$ m. A stationary bat sends out a sound signal and receives the same signal reflected from a moving moth at a frequency of 230.3 kHz. Calculate the magnitude of the velocity of the moth, assuming that the velocity is constant

Relevant Equations

Doppler Effect

I treat this question as two cases of Doppler effect.

(1) When the sound wave travels from bat to moth

Speed of sound = 222 x 1.5 = 333 m/s

Frequency received by moth:
$$f_1=\frac{333+v}{333}\times 222$$

(2) When the sound wave is reflected from moth back to bat

Frequency received by bat (moth as source and bat as observer):
$$f_2=\frac{333}{333-v}\times f_1$$
$$230.3=\frac{333}{333-v}\times \frac{333+v}{333}\times 222$$

Solving this equation, I get $v=6.1$ m/s but the answer key is 12 m/s

Where is my mistake?

Thanks

 

[Lnewqban]

Both, the stationary bat and the moving moth, receive sound of the same frequency.
That common frequency is greater than the frequency of the sound emitted by the bat.
It is the relative velocity of both creatures what reduces the distances between two consecutive crests of the sound wave, regardless which one is moving.

 

[Gavran]

Solving this equation, I get $v=6.1$ m/s but the answer key is 12 m/s

Where is my mistake?

Thanks

Your solution is okay. You can check it at https://en.wikipedia.org/wiki/Doppler_effect#Radar.

 

[songoku]

Both, the stationary bat and the moving moth, receive sound of the same frequency.

You mean both bat and moth will detect frequency of 230.3 kHz?

 

[haruspex]

Both, the stationary bat and the moving moth, receive sound of the same frequency.

Not so.

 

[haruspex]

the answer key is 12 m/s

Well, it doesn’t say the moth is flying directly towards the bat. Perhaps it was supposed to tell you it was flying at an angle of 60° to the direct route?

 

[songoku]

Well, it doesn’t say the moth is flying directly towards the bat. Perhaps it was supposed to tell you it was flying at an angle of 60° to the direct route?

How can we get this information from the question?

 

[Steve4Physics]

How can we get this information from the question?

Assuming that's not rhetorical...

You can’t. But there’s a possibility that a 60° angle was accidentally omitted from the question - this would explain official answer.

What you can do, at the start of your answer, is to make clear any assumption(s) you need to make.

E.g. “No information is supplied about the direction of the moth’s velocity. I will assume that we need to find the magnitude of the radial component of the moth’s velocity.”

 

[songoku]

Thank you very much for all the explanations Lnewqban, Gavran, haruspex, Steve4Physics