A bat moving at 3.7 m/s is chasing a flying
insect. The bat emits a 36 kHz chirp and
receives back an echo at 36.79 kHz.
At what speed is the bat gaining on its
prey? Take the speed of sound in air to be
340 m/s.
To determine the speed at which the bat is gaining on its prey, we can use the Doppler effect equation. The Doppler effect describes the change in frequency of a wave due to the motion of the source or the observer.
The equation for the Doppler effect is as follows:
f' = f * (V + Vd) / (V + Vs)
Where:
f' is the observed frequency
f is the emitted frequency
V is the velocity of sound
Vd is the velocity of the detector
Vs is the velocity of the source
In this case, the emitted frequency (f) is given as 36 kHz, and the observed frequency (f') is given as 36.79 kHz. The velocity of sound (V) is 340 m/s.
The bat, which is the detector in this case, is moving at a speed of 3.7 m/s. The velocity of the source (Vs) would be the speed at which the prey is moving.
To calculate the speed at which the bat is gaining on its prey, we need to solve the equation for Vs:
Vs = (V * (f' - f)) / (f' + f) - Vd
Plugging in the given values:
Vs = (340 * (36.79 kHz - 36 kHz)) / (36.79 kHz + 36 kHz) - 3.7 m/s
Calculating this expression will give us the speed at which the bat is gaining on its prey.