Aimed projectile launch angle

aimed_projectile_launch_angle.txt

An expression for the launch angle of a parabolic projectile which coincides with a target position vector (x, y) at some 
point in its trajectory.

The parametric coordinates of a parabolic projectile launched from the origin are
  
    x = uct
    
    y = ust + at² / 2
    
where 

    u is the initial velocity
    a is the acceleration
    t is the time since launch
    θ is the launch angle
    s = sin(θ)
    c = cos(θ)

Eliminating t we get

    y = x tan(θ) + ax²/(2u²c²)
    
      = x tan(θ) + (tan²(θ) + 1)ax²/(2u²)
      
Thus

    0 = (ax²/(2u²)) tan²(θ) + x tan(θ) + (ax²/(2u²)) - y

This is a quadratic equation in tan(θ) which can be solved for tan(θ)

Let 

    A = (ax²/(2u²))
    B = x
    C = A - y
    
Then

    0 = A tan²(θ) + B tan(θ) + C
    
and 

    θ = arctan( -B±√(B² - 4AC) )
              ( -------------- )
              (       2A       )