A problem for physics-major types.
At the instant of time shown, a particle has the velocity vector shown. Find the constant acceleration vector that brings the particle to the given landing point with the smallest possible speed of impact.
Introduce any variables you like.
(Note, this problem has nothing to do with gravity or any other forces. Alternatively, if you prefer to think of it this way instead, you might say that there was initially no gravity or any other force, and your task is to create a uniform force field of your own.)
Given any initial position vector, and given any initial velocity vector, does there always exist a constant acceleration vector such that the particle passes through the origin?