# The frequency of the train whistle is 445 Hz when the train is approaching, and 422 Hz when the train is receding. Calculate the speed of the train.I know to use the formula F0= Fs (v+- vo/...

The frequency of the train whistle is 445 Hz when the train is approaching, and 422 Hz when the train is receding. Calculate the speed of the train.

I know to use the formula F0= Fs (v+- vo/ v+-vs), but I just do not understand how to apply this formula correctly. Thank you so much for your help!!!

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### 1 Answer

When two objects are moving in a medium relative to one other, the frequency of sound emitted by one as measured by the other is different from the original frequency emitted. This is known as the Doppler Effect.

If the frequency of the signal being emitted by the source is f and the frequency of the signal as measured by the observer is f_m, the two are related with the relative speed of the two by the following equation

`f_m = f*((c + v_r)/(c + v_s))` where `v_s` and `v_r` refer to the velocity of the source and the receiver with respect to the medium and c is the velocity of the signal in the medium.

The speed of sound in air is 340.29 m/s.

Here, the speed of the observer is 0 and v_s is the speed of the train. f is the frequency of the sound emitted by the whistle as measured on the train.

`422 = f*((340.29 + 0)/(340 + v_s))` ...(1)

`445 = f*((340.29 + 0)/(340 - v_s))` ...(2)

Dividing the two gives

`422/445 = (340.29 - v_s)/(340.29 + v_s)`

=> `340.29*422 + 422*v_s = 340.29*445 - 445*v_s`

=> `v_s*867 = 7826.67`

=> `v_s = 9.027`

**The speed of the train is 9.027 m/s**

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