A train sounds its whistle as it approaches and crosses a level crossing. An observer at the crossing measures a frequency of 219 Hz as the train approaches and a frequency of 184 Hz as it leaves. If the speed of sound is taken to be `340m//s`, find the speed of the train and the frequency of its whistle

Explanation

When a whistling train approaches to rest observer, ` S overset( v_(s))(rarr) , O `
`v'= [ (v)/(v-v_(s))]v` ...(1)
When a whistling train away from rest observer, ` overset(v_(0))(rarr) ` S
`v^(n) [ ( v)/( v+ v_(s))]v` ...(2)
Here `v' = 219 Hz, v^(n) = 184Hz, v = 340 m//s `
`((1))/((2))implies (v')/(v'')=((v+v_(s)))/((v-v_(s)))`
`( 219)/(184) = ( 340+v_(s))/(340-v_(s))`
`219(340-v_(s))=184( 340 +v_(s))`
`219 xx 340 - 219v_(s) = 184xx 340 + 184v_(s)`
` 403 v_(s) = 35 xx 340`
`:.` Velocity of train, `v_(s) = 29.5 m //s `
Frequency of whistle , `v= v' xx[ ( v- v_(s))/(v) ] `
`= 219 xx [ ( 340 - 29.5)/(340)]`
`= 199.98`
`:. v = 200 H z`