Standing at a crosswalk, you hear a frequency of 560 Hz from the siren of an approaching ambulance, After the ambulance passes, the observed frequency of the siren is 480 Hz. Determine the ambulance's speed from these observations. Speed of sound `=343(m)/(s)`.

Explanation

We can assume that an ambulance with its siren on is in a hurry to get somewhere, and is probably travelling between 30 and `150(km)/(k)`, depending on the driving conditions. We can use the equation for the Doppler effect to find the speed of the vehicle. Let `v_a` be the magnitude of the velocity of the ambulance. As it approaches, you hear the frequency,
`f'=((v)/(v_v_a))f=560Hz`
The negative sign appears because the source is moving towards the observer. The opposite sign with source velocity magnitude describes the ambulance moving away. As it recedes, the Doppler shifted frequency is
`f')=((v)/(v+v_0))f=480Hz`
Solving the second of these equaition for f and substituting into the other gives
`f'=((v)/(v-v_0))(v+v_a)/(v))f')`
or `f'v-f'v_a=vf''+v_af'')`
so the speed of the source is
`v_a=(v(f'-f'))/(f'+f')=((343(m)/(s))(560Hz-480Hz))/(560Hz+480Hz)=26.4(m)/(s)`
This seems like a reasonable speed (about `85(km)/(h)`) for an ambulance, unless the street is crowded or the vehicle is travelling on an open highway.