A person with perfect pitch sits on a bus bench listening to the 444 Hz horn of an approaching car. If the person detects a frequency of 480 Hz, how fast is the car moving?
F = ((V+Vr)/(V-Vs))*Fs = 480 Hz.
((343+0)/(343-Vs))*444 = 480
((343)/(343-Vs))*444 = 480
152292/(343-Vs) = 480
Cross multiply:
164640-480Vs = 152,292
-480Vs = 152,292-164,640 = -12,348
Vs = 25.725 m/s.
F = Frequency as heard by receiver(man).
V = Velocity of sound in air.
Vr = Velocity of the receiver(man).
Vs = Velocity of source(car).
Fs = Frequency of source.
To determine the speed of the car, we can use the Doppler effect equation. The Doppler effect describes the change in frequency of a wave (in this case, sound) due to the relative motion between the source of the wave (the car) and the observer (the person with perfect pitch).
The equation to calculate the Doppler effect is:
f' = f * (v + vr) / (v + vs)
Where:
f' is the observed frequency
f is the actual frequency emitted by the source (the horn, in this case), which is 444 Hz
v is the speed of sound, approximately 343 meters per second
vr is the velocity of the receiver (the person with perfect pitch), which is assumed to be zero since they are sitting still on the bench
vs is the velocity of the source (the car), which is what we need to find
Rearranging the equation to solve for vs, we have:
vs = (f' * v - f * v) / (f' - f)
Plugging in the given values:
f' = 480 Hz
f = 444 Hz
v = 343 m/s
vs = (480 * 343 - 444 * 343) / (480 - 444)
vs = (164,640 - 152,292) / 36
vs = 12,348 / 36
vs ≈ 342.7 m/s
Therefore, the speed of the car is approximately 342.7 meters per second.
To determine how fast the car is moving, we can use the Doppler effect equation. The Doppler effect explains how the perceived frequency of a sound wave changes when the source of the sound is in motion relative to the observer.
The equation for the Doppler effect can be written as:
f' = (v + vo)/(v - vs) * f
Where:
f' is the perceived frequency (480 Hz in this case),
v is the speed of sound in air (approximately 343 m/s),
vo is the velocity of the observer (0 since the person is sitting still),
vs is the velocity of the source (the speed of the car horn that we're trying to find),
and f is the actual frequency of the car horn (444 Hz).
Now, we can rearrange the equation to solve for vs, the velocity of the source:
vs = ((f' / f) - 1) * v
Plugging in the given values, we get:
vs = ((480 Hz / 444 Hz) - 1) * 343 m/s
Simplifying:
vs = 0.0811 * 343 m/s
Therefore, the estimated velocity of the car is approximately 27.84 m/s.