A bat can detect small objects, such as an insect, whose size is approximately equal to one wavelength of the sound the bat makes. If bats emit a chirp at a frequency of 50.0 kHz, and if the speed of sound in air is 340 m/s, what is the smallest insect a bat can detect?
The smallest insect a bat can detect is approximately 6.8 mm in size. This is calculated by dividing the speed of sound (340 m/s) by the frequency of the chirp (50.0 kHz), which gives us 6.8 mm.
To calculate the smallest insect a bat can detect, we need to determine the wavelength of the sound wave produced by the bat.
The formula for calculating wavelength is:
wavelength = speed of sound / frequency
Given:
Frequency (f) = 50.0 kHz = 50,000 Hz
Speed of sound (v) = 340 m/s
Plugging in the values into the formula, we get:
wavelength = 340 m/s / 50,000 Hz
Now, let's convert the frequency to hertz (Hz) by dividing by 1,000:
wavelength = 340 m/s / 50,000 Hz
= 0.0068 meters or 6.8 millimeters
Therefore, the smallest insect a bat can detect is approximately 6.8 millimeters in size.
To find the smallest insect a bat can detect, we need to determine the wavelength of the sound wave emitted by the bat and compare it to the size of the insect.
The speed of sound in air is given as 340 m/s, and the frequency of the bat's chirp is 50.0 kHz (which is 50,000 Hz).
The relationship between frequency (f), wavelength (λ), and speed (v) is given by the equation:
v = f * λ
Rearranging the equation to solve for wavelength:
λ = v / f
Substituting the given values:
λ = 340 m/s / 50,000 Hz
Now, let's calculate the wavelength:
λ = 0.0068 meters or 6.8 mm
Therefore, the smallest insect a bat can detect is approximately equal to the size of the wavelength, which is 6.8 mm. Any object smaller than this size will go undetected by the bat's echolocation system.