A particle is projected up an inclined plane of inclination `beta` at na elevation `prop` to the horizontal. Show that
(a) `tan prop = cot beta + 2 tan beta`, if the particle strikes the plane at right angles
(b) `tan prop = 2 tan beta`, if the particle strikes the plane horizontally.

Explanation

(a) Let be the time of flight from O to A
Then`t=(2u sin(alpha-beta))/(gcosbeta)`
Now we shall consider the motion of (ground) the particle along OA.
Here initial velocity = u cos `(alpha-beta)`
Final velocity = 0
(The particle strikes the plane A atright angles
Acceleration of the partide along the incline plane, due to gravity = -g sin`beta`
`0= u  cos (alpha - beta)- g sin beta t`
or t =`(u cos(alpha-beta))/(g sin beta)`
From equation (1) & (2)  `(2u sin (alpha-beta))/(g cos beta)`=`(ucos(alpha-beta))/(g sin beta)`
or, 2 tan `(alpha-beta)` = cot `beta`
or, `2((tan alpha-than beta)/(1+ ten alpha tan beta))-cot beta`
or , `2 tan alpha-2tan beta cot beta + tan alpha`
or, `tan alpha cot beta + 2 tan beta`
(b) When the particle strikes the plane, horizontally i.e. along the ground In this case, `t=(2usin(alpha-beta))/(g cos beta)`
and 0 = u sin `alpha- gt`
or u sin `alpha=g[(2usin(alpha-beta))/(g cos beta)]`
or, sin `alpha cos beta`= 2 [ sin `alpha cos beta- cos alpha sin beta]`
or `2 cos alpha sin  beta= sin alpha cos beta`
`implies 2 tan beta = tan alpha`