In the game I am developing, I have to calculate if my vehicle (1) which in the example is travelling north with a speed V, can reach its target (2). The example depict the problem from atop:

There are actually two possible scenarios: V is constant (resulting in trajectory 4, an arc of a circle) or the vehicle has the capacity to accelerate/decelerate (trajectory 3, an arc of a spiral).

I would like to know if there is a straightforward way to verify if the vehicle is able to reach its target (as opposed to overshooting it). I'm particularly interested in trajectory #3, as I the only thing I could think of is integrating the position of the vehicle over time.

EDIT: of course the vehicle has always the capacity to steer, but the steer radius vary with its speed (think to a maximum lateral g-force).

EDIT2: also notice that (as most of the vehicles in real life) there is a minimum steering radius for the in-game ones too).