A particle is projected at an angle `prop` with horizontal from the foot of a plane whose inclination to `beta` Show that it will strike the plane at right angles if `cotbeta=2tan(prop-beta)`

Explanation

Let u be the velocity of projection so that u cos `(prop-beta)` and u sin `(prop-beta)` are the initial velocities respectively parallel and perpendicular to the inclined plane. The acceleration in these two directions are `(–g sin beta)` and `(–g cosbeta)`. The initial component of velocity perpendicular to PQ is u sin `(prop-beta)` and the acceleration in this direction is `(–g cosbeta)`. If T is the time the particle takes to go from P to Q then in time T the space described in a direction perpendicular to PQ is zero.
`0=u sin (prop-beta). T-1/2g cos betaT^2`
`T=(2usin(prop-beta))/(gcosbeta)`

If the direction of motion at the instant when the particle hits the plane be perpendicular to the plane, then the velocity at that instant parallel to the plane must be zero.
`:. ucos(prop-beta)-g sin betaT=0`
`(ucos(prop-beta)/(gsinbeta)=T=(2usin(prop-beta)/(gcosbeta)`
`:. cosbeta=2tan(prop-beta)`