A source of sound emitting a 1200 Hz note travels along a straight line at a speed of `170 m s^-1`. A detector is placed at a distance of 200 m from the line of motion of the source. (a) Find the frequency of sound received by the detector at the instant when the source gets closest to it. (b) Find the distance between the source and the detector at the instant it detects the frequency 1200 Hz. Velocity of sound in air` = 340 m s^-1`.

Explanation

When source is nearest to `O` i.e. at `A`, detector will receive that frequency which was emitted by source sometime before, let when source is at `P`. Let sound takes time `t` to reach from `P` to `O`,
`P O = v t = 340 t, PA = v_(s) t = 170 t`
`cos theta = (P A)/(P O) = (170 t)/(340 t) = (1)/(2)`
Frequency received at `O`
`f' = f((v)/(v - v_(s) cos theta)) = 1200 ((340)/(340 - (170)/(2)))`
`= 1600 Hz`
(b) Detector will receive a frequency `1200 Hz`, this frequency was emitted by source when it was at `A`. Time taken by sound to move from `A` to `O`,
`t = (200)/(340) s`
In this time source moves a distance
`= v_(s) t = 170 xx (200)/(340) = 100 m = A Q`
i.e. source is at `Q`.
`S O = sqrt((AO)^(2) + (AQ)^(2)) = sqrt((200)^(2) + (100)^(2)`
`= 100 sqrt(5) m`