A police officer with good ears hears an approaching motorcycle. The engine sound he eats as the bike approached is 195 Hz.
a) Is the actual sound the bike makes higher or lower in pitch?
b) After the bike passes he hears 147 Hz as the perceived pitch. If it was a nice 20 degree Celsius day, how fast was the bike moving.
-------
for a) I think that the actual sound in higher in pitch.
for b) I found the speed in air and got 343.52 m/s as my answer. I am not sure how to continue the question from there.
a) To determine whether the actual sound the bike makes is higher or lower in pitch, we need to consider the Doppler effect. The Doppler effect is the change in frequency or pitch of a sound wave when the source or observer is in motion relative to each other. When the source of the sound is moving towards the observer, the perceived pitch or frequency of the sound increases. Conversely, when the source is moving away from the observer, the perceived pitch or frequency decreases.
In this case, the police officer hears a sound of 195 Hz as the bike approaches. Since the police officer has "good ears," he is able to accurately perceive the actual frequency of the sound. If this is the actual frequency of the bike's engine sound, it means that the observed frequency (what the police officer hears) is the same as the actual frequency. Therefore, the actual sound the bike makes has a pitch of 195 Hz, and it is NOT higher or lower in pitch.
b) To determine the speed of the motorcycle, we can use the formula for the Doppler effect:
f = (v + vd) / (v + vs) * f0
where:
f = perceived frequency
f0 = actual frequency
v = speed of sound in the medium (in this case, air)
vd = velocity of the detector (the police officer)
vs = velocity of the source (the bike)
We are given f = 147 Hz and f0 = 195 Hz. The speed of sound in air at 20 degrees Celsius is approximately 343.2 m/s. The perceived frequency is lower than the actual frequency, which means that the source (the bike) must be moving away from the observer (the police officer).
To solve for vs, we rearrange the formula:
vs = (1 - f / f0) * v - vd
Substituting the given values, we have:
vs = (1 - 147 Hz / 195 Hz) * 343.2 m/s - 0 m/s (since the police officer is stationary)
Simplifying the equation:
vs = 0.25 * 343.2 m/s
vs ≈ 85.8 m/s
Therefore, the bike's speed is approximately 85.8 m/s.
You hear a high pitch as he comes toward you, lower as he leaves
So the actual frequency is lower than you hear.
In fact you know it is between 195 and 147
Fl = [(v + vl)/(v+vs) ] Fs
where
Fl = frequency heard
Fs = Frequency of source
v = v sound
vl = v listener
vs = v source
direction from listener TOWARD source is Positive
HERE:
You say v = 343.52
vl = police officer = 0
so
Fl = [v/(v+vs)]Fs
at first vs is negative, opposite from listener toward source
195 = [343.52/ (343.52 - vs)] Fs
147 = [343.52/(343.52 + vs)] Fs
eliminate Fs and solve for vs