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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 ...... $Hz$
$700$
$717$
$1000$
$679$
Solution
Sound from the source $P$ reaches to the observer at $\mathrm{Q}$ along the path $\mathrm{PQ}$. 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 $\mathrm{Q}$ is
$\mathrm{v}^{\prime}=\left(\frac{\mathrm{v}}{\mathrm{v}-\mathrm{v}_{\mathrm{s}} \cos \theta}\right) \mathrm{v}$
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}} $
${\approx 717 \mathrm{\,Hz}}$
