A particle is projected up an inclined plane of inclination `beta` at na elevation `prop` to the horizon. 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) `0 = u sin (prop - beta) t - (1)/(2) g cos beta t^2`
`(t = 2 u sin (prop - beta))/(g cos beta)`…(i)
`0 = u cos (prop - beta) - g sin beta t`
`rArr
t = (u cos (prop - beta))/(g cos beta)` …(ii)
From (i) and (ii), we get
`(2 sin (prop - beta))/(cos beta) = (cos(prop - beta))/(sin beta)`
`rArr 2 tan (prop - beta) = cot beta`
`rArr (2(tan prop - tan beta))/(1 + tan prop tan beta) = cot beta`
`rArr 2 tan prop - 2 tan beta = cot beta + tan prop`
`rArr tan prop = cot beta + 2 tan beta`
(b) `t = 2u sin (prop - beta)// g cos beta`
Also, `t = (u sin prop)/(g)`
`rArr (2 sin(prop - beta))/(cos beta) = sin prop`
`rArr 2 sin prop cos beta - 2 cos prop sin beta = sin prop cos beta`
`rArr sin prop cos beta = 2 cos prop sin beta`
`rArr tan prop = 2 tan beta`.
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