You are driving along a highway at 27.7 m/s when you hear a siren. You look in the rear-view mirror and see a police car approaching you from behind with a constant speed. The frequency of the siren that you hear is 1830. Hz. Right after the police car passes you, the frequency of the siren that you hear is 1760. Hz.
a) How fast was the police car moving?
b) You are so nervous after the police car passes you that you pull off the road and stop. Then you hear another siren, this time from an ambulance approaching from behind. The frequency of its siren that you hear is 1880. Hz. Once it passes, the frequency is 1680. Hz. What is the actual frequency of the ambulance’s siren?
((V+Vr)/(V+Vs))*Fs = 1760 Hz.
V = 343m/s = Velocity of sound in air.
Vr=27.7m/s=Velocity of receiver(motorist).
Vs = Velocity of source(Police car).
a. ((343+27.7)/(343+Vs))1830 = 1760
((370.7)/(343+Vs))1830 = 1760
678381/(343+Vs) = 1760
Cross multiply:
1760((343+Vs) = 678,381
343 + Vs = 385.44
Vs = 385.44 - 343 = 42.4 m/s = Velocity
of police car.
b. F = (1880+1680) /2 = 1780 Hz.
a) To find the speed of the police car, we can use the Doppler effect equation:
f' = ((v + vp) / (v - vs)) * f
Where:
- f' is the observed frequency
- v is the speed of sound in air (approximated to be 343 m/s)
- f is the actual frequency of the siren
- vp is the speed of the observer (your car)
- vs is the speed of the source (police car)
We are given:
- f' = 1760 Hz
- f = 1830 Hz
- vp = 27.7 m/s
- v = 343 m/s
Rearranging the equation for vs, we get:
vs = ((v + vp) / (f' / f) + 1)
Substituting in the given values:
vs = ((343 m/s + 27.7 m/s) / (1760 Hz / 1830 Hz) + 1)
vs = ((370.7 m/s) / (0.956) + 1)
vs = (370.7 m/s / 0.956 + 1)
vs = 390.07 m/s
Therefore, the speed of the police car was approximately 390.07 m/s.
b) Using the same Doppler effect equation as before, but now solving for f (the actual frequency of the ambulance's siren):
f' = ((v + va) / (v - vs)) * f
Where:
- f' is the observed frequency
- v is the speed of sound in air (approximated to be 343 m/s)
- f is the actual frequency of the siren
- vp is the speed of the observer (your car)
- vs is the speed of the source (ambulance)
- va is the speed of the ambulance
We are given:
- f' = 1680 Hz
- f = ? (what we need to find)
- vp = 0 m/s (since your car is stationary)
- v = 343 m/s
- vs = 343 m/s (assumed same as speed of sound since it's stationary)
Rearranging the equation for f, we get:
f = f' * ((v - vs) / (v + va))
Substituting in the given values:
f = 1680 Hz * ((343 m/s - 0 m/s) / (343 m/s + va))
We also know that after the ambulance passes, the observed frequency f' becomes 1880 Hz.
Rearranging the equation for va, we get:
va = -((v * (f - f')) / (f + f'))
Substituting in the given values:
va = -((343 m/s * (1830 Hz - 1760 Hz)) / (1830 Hz + 1760 Hz))
va = -((343 m/s * 70 Hz) / (3590 Hz))
va = -((343 m/s * 70 Hz) / (3590 Hz))
va ≈ -6.639 m/s
Therefore, the actual frequency of the ambulance's siren is approximately:
f = 1880 Hz * ((343 m/s - 343 m/s) / (343 m/s - (-6.639 m/s)))
f = 1880 Hz * (0 / 349.639 m/s)
f = 0 Hz
It appears there was an error in the calculations, and the result suggests that the actual frequency of the ambulance's siren is 0 Hz, which is not possible. Please double-check the given values and equations used to determine the actual frequency.
To solve this problem, we can use the concept of the Doppler effect. The Doppler effect explains the change in frequency of a sound wave when the source of the wave or the observer is moving.
a) Let's start with finding the speed of the police car. The frequency of the siren that you hear after the police car passes you is lower than the frequency you initially heard. This indicates that the police car was moving away from you.
The formula for the Doppler effect when the source is moving away is given by:
f' = f * (v + v₀) / (v + vs)
Where:
- f' is the observed frequency
- f is the actual frequency
- v is the speed of sound in air (approximately 343 m/s)
- v₀ is the speed of the observer (your speed)
- vs is the speed of the source (police car's speed)
Substituting the given values into the formula:
1760 Hz = 1830 Hz * (343 m/s + 27.7 m/s) / (343 m/s + vs)
Now, rearrange the equation and solve for vs (the speed of the police car):
vs = (1830 Hz - 1760 Hz) / (1760 Hz / (343 m/s + 27.7 m/s)) = 43.46 m/s
Therefore, the speed of the police car is approximately 43.46 m/s.
b) To determine the actual frequency of the ambulance's siren, we will use the same formula as in part a), but this time solving for f (the actual frequency):
1680 Hz = 1880 Hz * (343 m/s + 27.7 m/s) / (343 m/s + va)
Rearrange the equation and solve for va (the speed of the ambulance):
va = (1880 Hz - 1680 Hz) / (1680 Hz / (343 m/s + 27.7 m/s)) = 47.16 m/s
Therefore, the speed of the ambulance is approximately 47.16 m/s.
Now that we know the speeds of the police car and the ambulance, we can say that the actual frequency of the ambulance's siren is 1880 Hz.