The trajectory of a projectile in a vertical plane is `y=ax-bx^(2)`, where a and b are constantsn and x and y are respectively horizontal and vertical distances of the projectile from the point of projection. The maximum height height attained by the particle and the angle of projection form the horizontal are:

Explanation

`y = alpha x - beta x^(2)` (i)
Comparing it with the equation of trajectory of projectile,
we get
`y = x tan theta - (g x^(2))/(2 u^(2) cos^(2) theta)` (ii)
Comaparing the coefficients of `x` and `x^(2)`, we have
`tan theta = alpha rArr theta = tan^(-1) (alpha), theta`: angle of projection
`sin theta = (alpha)/(sqrt(1 + alpha^(2)))`
`cos theta = (1)/(sqrt(1 + alpha^(2)))`

`(g)/(2u^(2) cos^(2) theta) = beta rArr u^(2) = (g)/(2 cos^(2) theta.beta) = (g)/(2.(1)/(1 + alpha^(2)).beta)`
`u = sqrt((g(1 + alpha^(2)))/(2 beta)) , u`: velocity of projection
Maximum height `H_(max) = (u^(2) sin^(2) theta)/(2g)`
`= (g(1 + alpha^(2)))/(2g xx 2beta).(alpha^(2))/((1 + alpha^(2))) = (alpha^(2))/(4 beta)`
Range `R = (u^(2))/(g)(2 sin theta cos theta)`
`= (g(1 + alpha^(2)))/(g xx 2beta)(2.(alpha)/(sqrt(1 + alpha^(2))).(1)/(sqrt(1 + alpha^(2)))) = (alpha)/(beta)`
Alternatively
`y = alpha x - beta x^(2)`
For `y` to be maximum
`(d y)/(d x) = alpha - 2 beta x = 0 rArr x = (alpha)/(2 beta)`
`y_(max) = H_(max) = alpha((alpha)/(2 beta)) - beta((alpha)/(2 beta))^(2) = (alpha^(2))/(2 beta) - (alpha^(2))/(4 beta)`
`H_(max) = (alpha^(2))/(4 beta)`
When the height is maximum
`x = (R )/(2) rArr = 2 x = 2 (alpha)/(2 beta) = (alpha)/(beta)`
Another form of equation of trajectory:
The equation of trajectory of projectile
`y = tan theta - (g x^(2))/(2u^(2) cos^(2) theta)`
`= x tan theta [1 - (g x)/(2u^(2) cos^(2) theta tan theta)]`
`= x tan theta [1 - (g x)/(2u^(2) sin theta cos theta)]`
`= x tan theta [1 - (x)/((u^(2))/(g) (2 sin theta cos theta))]`
`y = x tan theta [1 - (x)/(R )], R`: Range
another form of equation of trajectory of projectile
`y^(2) = alpha x - beta x^(2) = alpha x(1 - (beta)/(alpha)x)` (i)
Comparing it with
`y = x tan theta (1 - (x)/(R ))` (ii)
`tan theta = alpha rArr theta = tan^(-1)(alpha)`
`(1)/(R ) = (beta)/(alpha) rArr = (alpha)/(beta)`