The figure shows the location of a source and detector at time t = 0 . The source and detector are moving with velocities v S = 5 i ^ m s - 1 and v D = 10 j ^ m s - 1 respectively. The frequency of si

Question

The figure shows the location of a source and detector at time $t = 0$ . The source and detector are moving with velocities $v_{S} = 5 \hat{i}$ $m s^{- 1}$ and $v_{D} = 10 \hat{j}$ $m s^{- 1}$ respectively. The frequency of signals received by the detector at the moment when the source crosses the origin is (the frequency of the source is 100 Hz. velocity of sound $330 m s^{- 1}$ )

TestHub · Accepted Answer

Time taken by 'S' to travel from S to 0 = $20 s$

In this time detector goes from $D_{1}$ to $D_{2}$

The signal received by the detector at $t = 20 s$ is not that produced by the source at 'O' but produced from a position $S_{1}$ prior to 'O'

$\frac{r}{v} = \frac{d}{v_{s}} \cos θ = \frac{d}{r} = \frac{v_{s}}{v} = \frac{5}{330} = \frac{1}{66}$

$\sin θ = \sqrt{1 - {(\frac{1}{66})}^{2}} \approx 1$

$\Rightarrow f_{D} = f (\frac{v - v_{0} \sin θ}{v - v_{s} \cos θ})$

$= 100 (\frac{330 - 10}{330 - \frac{5}{66}}) = 97 H z$

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