The problem can also be attacked using vectors; specifically, non-perpendicular components
Assume that the kicker is at the origin, the target is a distance $R$ at an angle θ, or at R∡θ.
The target is travelling at a speed V in a direction defined by the angle α; in other words, the target’s velocity is V∡α.
The kicker delivers the ball with a velocity S; we want to find if there is an angle β for the ball that will allow the ball to catch the target; in other words, find S∡β.
Assume first that one component of the kicked ball's velocity, S, matches the target's course and speed exactly, In other words, that component is V∡α. (Don't worry if V is greater than S!)
Since the ball’s movement is now matching the target’s movement exactly, the target will stay at the same distance and direction, R∡θ, relative to the ball. So the next thing we must do is add a velocity component with unknown size X towards the target; that is, with a velocity X∡θ. Again, these two components, V∡α, and X∡θ are not necessarily at right angles to each other. This velocity X will eventually cover the distance R, and the pass is a success! To finally solve the problem we need to calculate if a real, positive value for X exists, and what that value is. We can then find the angle β.
If we resolve these vectors into conventional x-y components and do some trig substitutions, we come up with the quadratic:
If this quadratic has no real roots, or two negative roots, there is no possible kick direction. If there is one positive solution or two positive solutions, then there is one or two possible kick directions.
The trig mentioned before also leads to the two equations:
$$S cos (β)=X cos (θ)+V cos (α)$$ and
$$S sin (β)=X sin (θ)+V sin (α)$$
and now that we know X, we can calculate the value(s) of β.
Oh, and R/X gives the travel time(s)...