An observer standing on a railway crossing receives frequencies of 2.2 kHz and 1.8 kHz when the train approaches and recedes from the observer. Find the velocity of the train in m/s.
[The speed of the sound in air is 300 m/s]

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Solution

$f^{†} = \frac{f_{0} (v \pm v_{o})}{(v \mp v_{s})} = \frac{f_{0} (v)}{v \mp v_{s}}$
Approaches: $2.2 = \frac{f_{0} (300)}{300 - v_{s}}$ (1)
Recedes: $1.8 = \frac{f_{0} (300)}{300 + v_{s}}$ (2)
Dividing (1) by (2),
$\frac{2. \cdot 2}{18} = \frac{300 + v_{s}}{300 - v_{s}} = \frac{4}{0.4} = \frac{600}{2 v_{s}}$
$= v_{s} = 30 m / s$

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