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The frequency of a steam train whistle as it approaches you is 538 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity)?

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For this problem, we need two Doppler formulas:

You are the stationary observer, and the source is moving:

(a) Toward:

f' = f/(1 - vs/v) = vf/(v - vs) (1)

(b) Away

f'' = f/(1 + vs/v) = vf/(v + vs) (2)

The last step is not so fancy, but very important:

Divide (1) by (2), we have:

f'/f'' = (v + vs)/(v - vs)

Therefore, vs = v (f' - f'')/(f' + f'') = 343 x (522 - 486)/(522 + 486) = 12.25m/s

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