The frequency of the sound, v = 1200 Hz. u = 170 m/, the speed of sound, V = 340 m/.
(a) The source is closest at O. {see the diagram below}.
But at this instant, the listener hears the sound produced at . The time taken t by the source in coming from to O is the same as the time taken by the sound in coming from * to P. Hence SP = Vt and SO = ut.
Hence cos α = SO/SP = ut/Vt = u/V. The sound is approaching the listener at * with a speed u' = u.cos α =u²/V.
Hence the frequency of sound heard by the listener v' = Vv/(V-u') = Vv/(V-u²/V) =V²v/(V²-u²)
→v' = 340²*1200/(340²-170²)
=340²*1200/(340+170)(340-170) Hz
=340²1200/(510170) Hz
=23401200/510 Hz
=221200/3 Hz
=4*400 Hz
=1600 Hz
(b) The detector will detect the original frequency of 1200 Hz when the sound produced at O reaches P, because when at P, the relative motion between O and P is zero. Time taken by the sound in traveling from O to P, t' = OP/V =200/340 =10/17 In this time the source reaches Q.
Hence, OQ = ut' = 17010/17 m =100 m.
The distance between the source and the detector when it detects the frequency of 1200 Hz = PQ.
→PQ² = PO²+OQ²
→PQ² =200²+100² =100²(2²+1)=100²*5
→PQ = 100√(5)
≈224 m
