Question

An projectile is fired uphill over ground with slope at an angle alpha to the horizontal.Find the direction in which it should be aimed to achieve the maximum range

Answer 1

Let's denote the distance as S and write the equations of the motion under the gravitational acceleration:
H = h₀ + v_oyt - gt²/2
S = v_ox t
Where vox = Vo cos(α), voy = Vo sin(α), ho = 0, H = 0.

0 = Vo sin(α)t - gt²/2; t =0; t = 2Vo sin(α) / g
S = Vo cos(α)t
S = 2Vo cos(α)Vo sin(α) / g = Vo sin (2α) / g , the maximum value of sin(2α) is 1 at 2α = 90, hence
α = 45 degrees.