Question

Two train whistles have identical frequencies of 1.64 102 Hz. When one train is at rest in the station and the other is moving nearby, a commuter standing on the station platform hears beats with a frequency of 4.00 beats/s when the whistles operate together. What are the two possible speeds that the moving train can have?

Answer 1

Vs = 6.73 m/s or Vs = 16.3 m/s

Step-by-step explanation:

frequency of the trains whistle (f) = 1.64 x 10^{2} Hz = 164 Hz

frequency of beats heard = 4 beats/s = 4 Hz

velocity of the stationary train (Vr) = 0

velocity of sound in air (V) = 343 m/s

velocity of the moving train (Vs) = ?

we can get the velocity of the moving train from the formula below

Fn = f x
`(V + Vr)/(V - Vs)` ...equation 1

where Fn = net frequency

• case one - assuming the train is approaching the station Fn = 164 + 4 = 168 Hz

substituting the known values into equation 1

168 = 164 x
`(343 + 0)/(343 - Vs)`

1.02 =
`(343 + 0)/(343 - Vs)`

Vs =
`343 - (343 + 0)/(1.02)`

Vs = 6.73 m/s

• case two - assuming the train is leaving the station Fn = 164 - 4 = 160 Hz

substituting the known values into equation 1

168 = 160 x
`(343 + 0)/(343 - Vs)`

1.05 =
`(343 + 0)/(343 - Vs)`

Vs =
`343 - (343 + 0)/(1.05)`

Vs = 16.3 m/s