How Fast is the Bat Flying Based on Doppler Effect and Beat Frequency?

[moronto]

Homework Statement

A bat emits a sound at a frequency of 37.0 kHz as it approaches a wall. The bat detects beats with a frequency of 832 Hz between the sound it emits and the echo bouncing from the wall. What is the speed of the bat? (The speed of sound is 335 m/s.)

Homework Equations

f_beat = |f1 - f2|
f1 = (1/1-vbat/v)f When the Bat is the source approaching the wall
f2 = (1+vbat/v)f1 When the Bat is the observer and the sound is echoed off the wall

The Attempt at a Solution

832 = |f1 - f2|
f1 = 832+f2
So essentially i have two equations, two unknowns
832+f2 = (1/1-vbat/335)37000
f2 = (1+vbat/335)832+f2
Worked to isolate f2 in both equations and equated them together, now this seems to be way too much work... numbers were huge and even when i worked it through the answer was wrong, What is wrong with my approach?

 

[anathema]

Your approach is correct, but you made a mistake in your equations. The correct equations should be:

f1 = (1/(1-vbat/335))37000
f2 = (1+vbat/335)f1

By plugging these into your calculator, you should get vbat = 17.5 m/s.

 

[Moetasim]

Your approach is on the right track, however, there are a few mistakes in your equations. First, the equation for the frequency of the beats should be f_beat = |f1 - f2|, not f_beat = |f1 + f2|. This is because the beats are caused by the difference in frequency between the sound emitted by the bat and the echo bouncing off the wall.

Second, in your equations for f1 and f2, you have included the frequency of the sound emitted by the bat (37.0 kHz) instead of the frequency of the beats (832 Hz). The correct equations should be f1 = (1/1-vbat/335)832 and f2 = (1+vbat/335)832.

Once you have corrected these mistakes, you should be able to solve for the speed of the bat using the equation for the speed of sound in air, v = fλ, where f is the frequency and λ is the wavelength. Remember to convert the frequency from kHz to Hz before plugging it into the equation. I hope this helps!