A car $P$ approaching a crossing at a speed of $10\, m/s$ sounds a horn of frequency $700\, Hz$ when $40\, m$ in front of the crossing. Speed of sound in air is $340\, m/s$. Another car $Q$ is at rest on a road which is perpendicular to the road on which car $P$ is reaching the crossing (see figure). The driver of car $Q$ hears the sound of the horn of car $P$ when he is $30\, m$ in front of the crossing. The apparent frequency heard by the driver of car $Q$ is .... $Hz$
Diffcult
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Sound from the source $P$ reaches to the observer at $\mathrm{Q}$ along the pth $\mathrm{PQ}$. Source $\mathrm{P}$ is approaching the crossing with velocity $\mathrm{v}_{\mathrm{s}}=10 \mathrm{m} / \mathrm{s}$
When the observer in car $Q$ hears the sound of the horn, the effective velocity of approach of the car $\mathrm{P}$ towards observer is $\mathrm{v}_{\mathrm{s}} \cos \theta.$
Thus, apparent frequency heard by the observer in car $Q$ is
${v^{\prime}=\left(\frac{v}{v-v_{s} \cos \theta}\right) v} $
Here ${\cos \theta=\frac{4}{5}}$
$\therefore $ $ v^{\prime}=\frac{340}{340-10 \times \frac{4}{5}} \times 700=\frac{340}{332} \times 700 $
$ =716.86 \mathrm{Hz} $
$ \approx 717 \mathrm{Hz} $
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