Based on frequencies, calculate how fast this vehicle is moving...?

Question

An ambulance with a siren emitting a whine at 1,600 Hz overtakes and passes a cyclist pedaling a bike at 2.44 m/s. After being passed, the cyclist hears a frequency of 1,590 Hz. How fast is the ambulance moving?

Explain your work. Thanks! :)

Answer 1

Simple application of the Doppler effect:
http://en.wikipedia.org/wiki/Doppler_effect

Fo = Fr(V/(V + Vd)) where

Fr = real frequency emitted by the ambulance
Fo = the frequency as observed by the cyclist
V = the speed of sound
Vd = the difference in speed between the source and the observer

You know Fr and Fo. You can get V easily (but see note 3 below), so you can solve for Vd. Given Vd and the velocity of the cyclist, you can compute the velocity of the ambulance.

Note that:

1. The "+" sign is there because the source is moving away from the observer. It would be "-" if the source were moving toward the cyclist.

2. This is a first-order approximation. You can separate the problem into siren vs stationary observer and stationary sound vs the cyclist, but since we are working with relatively slow speeds (compared with the velocity of sound in air) there is no need to do so.

3. The velocity of sound is affected by the pressure and temperature so that the answer will be significantly different at a high altitude city in winter and a tropical city in the summer (Denver vs. Rio, etc.)
http://en.wikipedia.org/wiki/Speed_of_sound