Two train whistles have identical frequencies of 180 Hz. When one train is at rest in the station and the other is moving nearby, a passenger standing on the station platform hears beats with a frequency of 2.00 beats/s when the whistles operate together. What are the two possible speeds and direction the moving train can have? speed of sound is 343 m/s.

Explanation

A Doppler shift in the frequency received from the moving tran is the source of the beats. We consider velocities of approach and of recession separately in the Doppler equation after we observes from our beat equation `f_b=|f_1-f_2|=|f-f'|` that the moving train must have an apparent frequency of either `f'=182Hz` of `f'=178Hz`. We let `v_t` represent the magnitude of the train's velocity. If the tran is moving away from the station the apparent frequency is 178 Hz lower, as described by ltbr. `f'=(v)/(v+v_t)f`
and the train is moving away at
`v_t=((f)/(f')-1)upsilon=(343(m)/(s))((180Hz)/(178Hz)-1)=3.85(m)/(s)`
If the train is pulling into the station, then the apprent frequency is 182 Hz. Again from the Doppler shift,
`f'=f(v)/(v-v_S)`
The train is approaching at
`v_S=upsilon(1-(f)/(f'))=(343(m)/(s))(1-(180Hz)/(182Hz))`
`=3.77(m)/(s)`
The Doppler effect does not measure distance, but it can be a remarkably sensitive measure of speed.