A bat moving at 6.5 m/s is chasing a flying
insect. The bat emits a 37 kHz chirp and
receives back an echo at 37.65 kHz.
At what speed is the bat gaining on its
prey? Take the speed of sound in air to be
336 m/s. Answer in units of m/s
To determine the speed at which the bat is gaining on its prey, we first need to calculate the frequency shift between the emitted chirp and the received echo.
The frequency shift, denoted as Δf, can be calculated using the formula:
Δf = f_r - f_e
where f_r is the received frequency and f_e is the emitted frequency.
In this case, the emitted frequency is 37 kHz and the received frequency is 37.65 kHz. Plugging in these values, we get:
Δf = 37.65 kHz - 37 kHz
Next, we need to convert the frequency shift into a velocity shift. This can be done using the Doppler effect formula:
Δf/f_e = v/c
where Δf is the frequency shift, f_e is the emitted frequency, v is the velocity shift, and c is the speed of sound.
In this case, we know the frequency shift (Δf) and the speed of sound (c). We want to find the velocity shift (v). Rearranging the formula, we get:
v = Δf/f_e * c
Plugging in the values, we have:
v = (Δf/f_e) * c = (0.65 kHz / 37 kHz) * 336 m/s = 5.888 m/s
Therefore, the bat is gaining on its prey at a speed of approximately 5.888 m/s.