A source of sound emitting a 1200 Hz note travels along a straight line at a speed of 170 m/s. A detector is…
A source of sound emitting a 1200 Hz note travels along a straight line at a speed of 170 m/s. A detector is placed at a distance of 200 m from the line of motion of the source. Find the frequency of sound received by the detector at the instant when the source gets closest to it.
(Take, speed of sound in air = $340\text{ ms}^{-1}$)
1600 Hz
1000 Hz
1700 Hz
1200 Hz
Solution
When source is nearest to $O$, i.e. at $A$, detector will receive that frequency which was emitted by source sometime before, assume when source is at $P$. Let sound takes time $t$ to reach from $P$ to $O$.
[Diagram showing a source moving horizontally along line $SQ$ at $v_s = 170 \text{ m/s}$ past points $P$ and $A$, with detector $O$ located $200 \text{ m}$ directly below point $A$, forming right triangle $PAO$ with angle $\theta$ at $P$.]
$PO = vt = 340t, \quad PA = v_s t = 170t$
$\cos \theta = \frac{PA}{PO} = \frac{170t}{340t} = \frac{1}{2}$
Frequency received at $O$,
$f' = f \left(\frac{v}{v - v_s \cos \theta}\right) = 1200 \left(\frac{340}{340 - \frac{170}{2}}\right) = 1600 \text{ Hz}$