My full question is: There is a movable point $P$ in a flat 2D space. The magnitude of its acceleration cannot exceed $a_{max}$. Given its initial position to be at $(0,0)$, and its initial velocity to be $(v_x,0)$ at $t=0$, an area $A$ evolving with time consists of all possible positions $P$ can be at time $t$. Describe $A$ or its boundary with respect to time, using $a_{max}$ and $v_x$.
I guess the first step is to be given another point $(x,y)$, and calculate the fastest way to go there under the acceleration limit, then calculate the minimum time required. I don't know how to do that. Partial or full solutions are both appreciated.
Let's define a new coord system in which the origin is moving to the right with velocity $(v_x, 0)$ in the old coordinate system. In this NEW system, the question is the same as the original, but the initial velocity is zero.
The set of reachable positions, at time $t$ in the new coordinate system, consists of the interior of a disk of radius $\frac{a}{2} t^2$ (by ... well... integrating).
So in the old coordinate system, the set of reachable positions is the disk of radius $r = \frac{a}{2}t^2$, with center at $(tv_x, 0)$.