First slide

Doppler

Question

A train is moving with a constant speed u along a circular track. A siren in its engine is emitting a sound of frequency v. If u = v/10, where v is the speed of sound, the apparent frequency of sound as heard by a passenger at the rear end of the train is

Difficult

A

$\frac{9 v}{10}$
B

$\frac{11 v}{10}$
C

$\frac{9 v}{11}$
D

$v$

Solution

Refer to Fig.
The components of ${\vec{u}}_{0} and {\vec{u}}_{s}$ along the direction of the velocity of sound are $u_{0} cosθ$ and $u_{s} cosθ$ given $u_{0} = u_{s} = u$ . Thus, the source is receding from the observer with a speed $u \cos θ$ and the observer is approaching the source with the same speed $u \cos θ$
$∴$ Apparent frequency $v' = v (\frac{v + ucosθ}{v + ucosθ}) = v$
Hence the correct chose is (d)

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