A particle is projected up an inclines plane. Plane is inclined at an angle `theta` with horizontal and particle is projected at an angle `alpha` with horizontal. If particle strikes the plane horizontally prove that `tan alpha=2 tan theta`

Explanation

We know time of flight
`T=(2u sin (alpha + theta))/(g cos theta)`
Considering vertical motion
`u=v sin alpha`
`a=-g`
`v=theta`
`:. T=(u sin alpha)/g=(2u sin (alpha - theta))/(g cos theta)`
`sin alpha cos theta = 2 sin alpha cos theta -2 cos alpha sin theta`
`2 cos alpha sin theta= sin alpha cos theta`
`2 tan theta= tan alpha`