But wait. Why are you giving examples in 2-space? Isn't this 3-space?
I attempted the problem on my own using vector analysis (Velocity is a vector quantity that can be broken up into 3 subsections, x, y, and z by using trigonometry and two angles. Yes, two angles in 3-space. One to determine how far above the ground and how far in the x-direction and another to determine how far off of the xy plane it is (z).)
I got extremely complicated formulas that did not effectively isolate theta and alpha (my two angles). I had 3 equations with 3 unknowns, though. Here's what I had, maybe someone else can extrapolate a bit more:
T = Target
R = projectile
V = Target velocity
S = projectile velocity
[] = anything within there is a subscript.
t = time
Target directions are given in the problem statement, so the only unknowns are time, and theta and alpha.
T
- = T[x[0]] + Vcos(theta[Target direction])t
T[y] = T[y[0]] + Vsin(theta[Target direction])t
T[z] = T[z[0]] + Vcos(alpha[Target direction])t
R
R[y] = 0 + Ssin(theta)t
R[z] = 0 + Scos(alpha)t
In my analysis, since there is no acceleration, basic kinematic equations can be used. The equation I used is x[t] = x[0] + vt + 1/2at^2. a = 0 so it's simply x[t] = x[0] + vt. Since V is a velocity magnitude, in order to find the direction in a certain axis direction basic trig must be used to find exactly how much of V works in that direction.
Also, I "moved" my axis system so that the origin was located at where the projectile was fired. It's a simple calculation, you only subtract the particles initial position from the targets initial position and set the particle at the origin. It simplifies the algebra a bit, yet does not yield a fully simplified answer using the tools that I currently know.
So, after that little note you can set those six equations equal (In their respective axes), solve for t, substitute, and you will have 2 equations with only theta and alpha as unknowns. It's only complicated because simplifying trig functions isn't as easy as factoring out common factors in algebraic functions.