# Find direction of acceleration to arrive at a point with no velocity

A rocket ship is at a position p in 2D Euclidean geometry/space travelling at velocity v. It is capable of accelerating at a constant acceleration a, but is capable of providing this acceleration in any direction. Which directions should the ship point in over time in order to arrive at another point d as soon as possible? The ship must hit zero velocity at the same time as it arrives at d.

There is no limit on the speed of the ship's rotation, and there is no friction. The ship can be approximated as a point mass (in case that's somehow relevant). The points do not move.

As my application will be performing this calculation every tick, I only really need to know which direction the ship points in at the first instant.

It would also be useful (but certainly not necessary) to know how this plays out in elliptic geometry/space (again in 2D- a.k.a. the surface of a sphere).

EDIT: There are no external forces acting on the ship.

Are you implying that the engine has only two speeds, on and off, so you get either |acceleration| = a or nothing?

My intuition is that, not accounting for gravity (you are well away from the solar system) the fastest path is a straight line $P_0$ to $P_1$. You accelerate for the first half of the path, then turn 180 degrees and decelerate at the same a for the second half, arriving with velocity = 0.

The reasoning, which you should support with a little calculation, is that follwing a path f(t) the velocity and acceleration in the $P_0P_1$ direction will be their projections of the tangent vector $f'(t)$ on $P_0P_1$ and therefore less than could be attained in the straight line.

On the surface of a sphere the fastest path should be the great circle, again to calculate.

• I think that would work if the rocket were originally at rest, and something similar might work if the initial velocity was heading in the direction of the point P. Otherwise it would be much more complicated.
– WW1
Mar 11 '17 at 22:11
• There is no minimum acceleration, anything less than a certain cap is acceptable. And as WW1 pointed out, that only works of the ship starts at rest, which it does not. You were correct in assuming that gravity was negligible. Mar 12 '17 at 17:43
• This will take some calculation which I have no time to do now, but my intuition is that is v is small compared to a you get essentially the straight line anyway; if v is large compared to a the fastest way is to point in the direction of $P1$ so as to get the maximum acceleration and speed in the desired direction. Then at the appropriate point -- an interesting calculation -- you turn around and decelerate. At least in the acceleration phase you'll spiral in, like a satellite orbit decaying. Mar 14 '17 at 4:41