I have a spaceship moving in a 2D plane and wish to intercept another body which is also moving in the same plane. To do this I need to find a series of accelerations over a series of times that will result in the relative velocity = zero and also the relative distance = 0. Maximum acceleration is fixed but there is no upper limit on speed.
To simplify the problem, I first calculate the relative velocity in polar coordinates, then switch the local axes so that in local coordinates, the spaceship is positioned at the origin with a known velocity and I want it to arrive at a static target (at angle = 0) then stop.
Calculating this in 1D is quite simple and can be solved by equating the various kinematic equations. This can be solved in both parallel and perpendicular directions separately, but even though we know that $a,max^2 = a,ll^2 + a,T^2$, I cannot figure out how to combine the two or calculate a suitable ratio of accelerations even approximately.
Just to summarise, we know distance, angle, relative velocity and maximum acceleration, and are looking to find a series of 2 (or more) accelerations over an equal number of time periods. The end state is the two bodies sharing the same coordinates with no relative velocity.
This has had me stumped for a week so I would be very grateful for a solution!
You could first do the manouver in the $x$-direction to make sure that the $x$-coordinate is constant and equal to the $x$-coordinate of the target (assuming you are in a reference frame where the target is stationary). This is the same problem as the one-dimensional problem. Once that you have done that you do the same in the $y$-direction. Then you are done.