The frequency of an ambulance's siren is 700 Hz.
If a pedestrian on the side of the road hears the siren at 756 Hz, approximately how fast and in what direction is the ambulance moving?
a- There is not enough information
b- 25 km/h, away from the pedestrian
c- 90 km/h, towards the pedestrian
d- 90 km/h, away from the pedestrian
e- 25 km/h, towards the pedestrian
To solve this problem, we need to apply the Doppler effect formula for sound. The Doppler effect is the change in frequency or wavelength of a wave for an observer moving relative to the source of the wave.
The formula for the Doppler effect is given by:
f' = f * (v + vobs) / (v + vs),
where f' is the observed frequency, f is the source frequency, v is the speed of sound in air (approximately 343 m/s), vobs is the velocity of the observer, and vs is the velocity of the source.
Given that the observed frequency f' is 756 Hz and the source frequency f is 700 Hz, we can rearrange the formula to solve for the velocity of the ambulance (vs):
vs = (f' - f) * (v + vobs) / (f + v)
Substituting the known values into the equation:
vs = (756 Hz - 700 Hz) * (343 m/s + vobs) / (700 Hz + 343 m/s)
Simplifying the equation gives:
vs = 56 Hz * (343 m/s + vobs) / (1043 Hz)
To determine the direction and speed of the ambulance, we need to compare the observed frequency (756 Hz) to the source frequency (700 Hz).
If the frequency increases (observed frequency is greater than the source frequency), it means the ambulance is moving towards the pedestrian. On the other hand, if the frequency decreases (observed frequency is less than the source frequency), it means the ambulance is moving away from the pedestrian.
In this case, the observed frequency is higher than the source frequency, indicating that the ambulance is moving towards the pedestrian.
As for the speed, we can determine it by analyzing the equation above. Since we have no information on the velocity of the observer (vobs), we cannot calculate the exact speed but we can make a general estimation based on the answer choices given.
Based on the available answer choices, c- 90 km/h, towards the pedestrian is the most likely option as it suggests the ambulance is moving towards the pedestrian at a relatively high speed.