[Jack Simth]

I'm not so much solving for T as making a connection between dimensions out of it. It gets around having to play with sine and cosine.

Ax and Ay are related by the equation: Amax = sqrt(Ax^2+Ay^2) (for two dimensions, to get soonest interception for a "ram target" scenario). Amax is a constant. You now have three equations:

(1/2)(Ax - Ax)(T^2)+(Vx - Vx)T+(Sx- Sx) = 0
(1/2)(Ay - Ay)(T^2)+(Vy - Vy)T+(Sy- Sy) = 0
Amax = sqrt(Ax^2+Ay^2)

We have many symbols:
T: Unknown, time of interception.
Ax: Unknown, acceleration in X for the chaser.
Ax: Known, acceleration in X for the chased.
Vx: Known, starting velocity in X for the chaser.
Vx: known, starting velocity in X for the chased.
Sx: known, starting position in X for the chaser.
Sx: known, starting position in X for the chased.
Ay: unknown, acceleration in y for the chaser.
Ay: known, acceleration in y for chased.
Vy: known, starting velocity in y for chaser.
Vy: known, starting velocity in y for chased.
Sy: known, starting position in y for chaser.
Sy: known, starting position in y for chased.
Amax: known, based on the pursuer - maximum acceleration used(which is the acceleration you'll normally want to use for the chaser, and will likely be some arbitrary % of "true max").

However, we've only got three unknown values: T, Ax, and Ay. We have three equations that relate them (if we had three dimensions, we'd have a fourth unknown, but a fourth equation to go with it, so they don't matter).

I'm a little too tired to do the brute-force logic manipulations on this at the moment (overtime at work) but at this point you should be looking at nothing more than time-consuming symbol manipulation and replacement set to solve for all of them. You do them all at essentially the same time. Manipulate the equations until you have each in terms of the other three, then replace. Eventually, you'll get one unknown on one side and a (somewhat big) equation of knowns on the other - at which point you _finally_ apply actual math to it (or rather, tell the computer to do so). Then that value is "known". Once that value is known, you plug it into the spots on the other equations, and repeat the process.