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

Answer:

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

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 $\frac{V + Vr}{V - Vs}$ ...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 $\frac{343 + 0}{343 - Vs}$

1.02 = $\frac{343 + 0}{343 - Vs}$

Vs = $343 - \frac{343 + 0}{1.02}$

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 $\frac{343 + 0}{343 - Vs}$

1.05 = $\frac{343 + 0}{343 - Vs}$

Vs = $343 - \frac{343 + 0}{1.05}$

Vs = 16.3 m/s