An ambulance with a siren emitting a whine at 1600 Hz overtaken and passes a cyclist pedaling a bike at 2.44 m/s. After being passed, the cyclist hears a frequency of 1590 Hz. How fast is the ambulance moving?
The given frequencies appear to be too
close together; and the answer is negative. Please check.
To determine the speed of the ambulance, we need to utilize the Doppler effect equation. The equation relates the observed frequency to the source frequency and the relative velocity between the observer and the source.
The Doppler effect equation is given by:
f' = f * (v + vo) / (v + vs)
Where:
- f' is the frequency observed by the observer (cyclist)
- f is the frequency emitted by the source (ambulance)
- v is the speed of sound in air (assumed to be 343 m/s)
- vo is the velocity of the observer (cyclist)
- vs is the velocity of the source (ambulance)
Let's plug in the given values into the equation and solve for the velocity of the ambulance.
f' = 1590 Hz (observed frequency)
f = 1600 Hz (source frequency)
v = 343 m/s (speed of sound in air)
vo = 2.44 m/s (velocity of the observer)
vs = ?
1590 = 1600 * (343 + 2.44) / (343 + vs)
Simplifying the equation:
1590 = 1600 * (345.44) / (343 + vs)
Multiplying both sides by (343 + vs):
1590 * (343 + vs) = 1600 * 345.44
544770 + 1590vs = 552704
1590vs = 7934
vs = 7934 / 1590
vs ≈ 4.99 m/s
Therefore, the velocity of the ambulance is approximately 4.99 m/s.