A bird is flying directly toward a stationary bird-watcher and emits a frequency of 1390 Hz. The bird-watcher, however, hears a frequency of 1430 Hz. What is the speed of the bird, expressed as a percentage of the speed of sound?
f =f₀/(1-v/u)
f₀=1390 Hz
f=1430 Hz
v/u=(f-f₀)/f= (1430-1390) /1430=2.8•10⁻² m/s
v/u = 2.8%
To determine the speed of the bird relative to the speed of sound, we can use the formula for the Doppler effect:
Δf/f = v/c
Where:
Δf is the change in frequency (observed frequency minus emitted frequency)
f is the emitted frequency
v is the velocity of the source (bird)
c is the speed of sound
Given:
Δf = 1430 Hz - 1390 Hz = 40 Hz
f = 1390 Hz
Substituting the values into the equation, we have:
40 Hz / 1390 Hz = v / c
To find the speed of the bird (v) as a percentage of the speed of sound (c), we can rearrange and solve for v:
v = (40 Hz / 1390 Hz) * c
The speed of sound (c) can vary depending on atmospheric conditions, but we can assume a typical value of approximately 343 meters per second (or 34300 cm/s).
v = (40 Hz / 1390 Hz) * 34300 cm/s
Calculating this expression:
v ≈ 988.53 cm/s
To express this speed as a percentage of the speed of sound (c):
(v / c) * 100%
Substituting the values:
(988.53 cm/s / 34300 cm/s) * 100%
Calculating this expression:
≈ 2.88%
Therefore, the speed of the bird, expressed as a percentage of the speed of sound, is approximately 2.88%.
To solve this problem, we need to apply the Doppler effect formula. The Doppler effect describes the change in frequency of a wave (in this case, sound waves) due to the relative motion of the source and the observer.
The formula for the apparent frequency heard by the observer is:
f' = f * (v + v₀) / (v + vᵥ)
Where:
- f' is the observed frequency (1430 Hz in this case)
- f is the emitted frequency by the source (1390 Hz in this case)
- v is the speed of sound in the medium
- v₀ is the speed of the observer (the bird-watcher)
- vᵥ is the speed of the source (the bird)
Since the bird is flying directly toward the bird-watcher, its speed can be considered positive.
To find the speed of the bird, expressed as a percentage of the speed of sound, we can rearrange the formula as follows:
vᵥ = (f - f') * (v + v₀) / (f + f')
To calculate the speed of the bird, we need to know the speed of sound (v) and the speed of the observer (v₀).
Once we have these values, we can substitute them into the equation to find the speed of the bird-vᵥ. Then, we divide it by the speed of sound-v to express the bird's speed as a percentage of the speed of sound.
Please provide the values of the speed of sound (v) and the speed of the observer (v₀), and I'll calculate the speed of the bird for you.