Question

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

Answer 1

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

Let's denote the distance as S and write the equations of the motion under the gravitational acceleration:
H = h0 + Vy*t - gt2/2
S = Vx*t.
Here Vx = V0*cos(α), Vy = V0*sin(α), h0 = 0, H = 0.

0& =& Vo sin(α)t - gt²/2; &
t =0;& t = 2V0*sin(α)/g
S = V0*cos(α)t &
S& = 2V0*cos(α)V0*sin(α)/g = V0*sin(2α)/ g , the maximum value of& sin(2α) is 1 at 2α = 90, hence α = 45 degrees.