A train approaching a hill at a speed of `40 km//hr` sounds a whistle of frequency `580 Hz` when it is at a distance of `1km` from a hill. A wind with a speed of `40km//hr` is blowing in the direction of motion of the train Find
(i) the frequency of the whistle as heard by an observer on the hill,
(ii) the distance from the hill at which the echo from the hill is heard by the driver and its frequency.
(Velocity of sound in air `= 1, 200 km//hr`)

Explanation

(a) Speed of sound in air `= 1200 km//hr`
Speed of source `v_(s) = 40 km//hr`
Speed of wind `w = 40 km//hr`
Frequency emitted by source `f = 580 Hz`

Frequency received at hill
`f' = f((v+w-v_(0))/(v+w-v_(s)))`
`= 580 ((1200 + 40 -0)/(1200 + 40-40))`
`= 599.33 Hz`
(b) Let echo will be heard by driver when train is at `A` i.e., at distance `x` from hill.
Time taken by train to reach `A`,
`t_(1) = (SA)/(v_(s)) = (1-x)/(v_(s))`
Time taken by echo to reach `A`,
`t_(2) = (SO)/(v+w) + (OA)/(v-w) = (1)/(v+w) + (x)/(v-w)`
`t_(1) = t_(2)`
`(1-x)/(v_(s)) = (1)/(v+w)+(x)/(v-w)`
`(1-x)/(49) = (1)/(1200+40)+(x)/(1200-40)`
Solving, `x = (29)/(30)km = 933.3m`
Frequency of reflected sound heard by driver
Hill will act as a source of frequency `f'` and driver as an observer.
`{:(rarr v_(o)=40 km//hr,,v_(s)=0 larr),(overset(*)(O) " " rarr w,," "overset(*)(S)),(" "v larr,,):}`
Frequency of reflected sound
`f_( r) = f'((v-w+v_(O))/(v-w-v_(O)))`
`= (599.33)((1200-40+40)/(1200-40-0))`
`= 620 Hz`