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See we need to find the time at which initial velocity vector is perpendicular to velocity vector at that time ,
initially the angle is theta ( I'm considering the angle as @ )
resolving initial velocity vector we get, usin@j^ + ucos@i^
Now at any time t velocity vector of projectile is given by formula,
V vector = ucos@i^ + (usin@ - gt)j^
We have initial and final required velocity vector,
since it is given they are perpendicular their dot product must be 0
⇒ u^2cos^2@ + u^2sin^2@ - gtusin@ = 0
= u^2 - gtusin@ = 0
⇒ t= u/gsin@
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