A bat flying at a speed of 5 m/s chases an insect flying in the same direction. If the bat emits a 40 kHz sound and hears the eco of 40 kHz. What is the speed of the insect? Consider the speed of sound is 340 m/s
if there's no Doppler shift, then they must be going the same speed.
To find the speed of the insect, we can apply the Doppler effect formula. The Doppler effect describes the change in frequency of a wave (in this case, sound) due to the relative motion between the source (bat) and the observer (bat hearing its own echo).
The formula for the Doppler effect in this situation, where the source and observer are moving towards each other, is:
f' = (v + vo) / (v + vs) * f
Where:
f' = observed frequency (echo frequency) = 40 kHz
v = speed of sound = 340 m/s
vo = speed of the observer (bat) = 5 m/s
vs = speed of the source (insect) = unknown
f = emitted frequency = 40 kHz
Now, we can rearrange the formula to solve for the speed of the source (insect):
f' = (v + vo) / (v + vs) * f
Rearranging the formula:
(vs + v) / (v + vo) = f / f'
Substituting the given values:
(vs + 340) / (340 + 5) = 40,000 / 40,000
(vs + 340) / 345 = 1
Cross-multiplying:
vs + 340 = 345
vs = 345 - 340
vs = 5 m/s
Therefore, the speed of the insect is 5 m/s.