An aeroplane flies along a straight path `A and B` and returns back again. The distance between `A and B` is `l` and the aeroplane maintains the constant speed `v` w.r.t. wind.
There is a steady wind with a speed `u` at an angle `theta` with line `A B`. Determine the expression for the total time of the trip.
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Explanation

Suppose the plane is oriented at an angle `prop` w.r.t. line `AB`, while the plane is moving from `A` to `B` :
Velocity of plane along
`AB = v cos prop - u cos theta`, and for no - drift from line `AB`
`v sin prop = u sin theta`
`rArr sin prop = (u sin theta)/(v)`
Time taken from `A` to `B ` : `t_(A B) = (l)/(v cos prop - u cos theta)`
Suppose plane is oriented at an angle `prop` w.r.t. line `AB` while the plane is moving from `B` to `A` :
velocity of plane along `BA = v cos prop + u cos`
`theta` and for no drift from line `AB`.
`v sin prop = u sin theta`
`rArr sin prop = (u sin theta)/(v) rArr prop = prop'`
Time taken from `B` to `A` :
`t_(B A) = (l)/(v cos prop + u cos theta)`
Total time taken `= t_(A B) + t_(B A)`
`= (l)/(vcosalpha - cosalpha) + (l)/(vcosalpha + ucosalpha)`
=`(2vlcosalpha)/(v^2 cos^2 prop + u^2 cos^2 theta) = (2 v l sqrt(1 - (u^2 sin^2 theta)/(v^2)))/(v^2 - u^2)`.
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