physics - How to implement deceleration and stopping over a certain distance

So I have a a character, say a spaceship. It needs to move distance R, and in direction T (Theta). So say if The object is at (0,0), and it needs to get to (4,3), it has R = 5 and T = about 36 degrees.

Essentially I know where the endpoint is, and the distance away. I want my spaceship to start decelerating when R becomes 8, and the object come to a halt when it reaches the location.

How can I do this. (Kinematics in answers is welcome)

Answer

Here are 3 standard (pre-calculus) kinematics formulae covering the case of constant acceleration, each with one of the unknowns (t, v, or _d) eliminated:

1. v^2 = u^2 + 2 a d


2. v = u + a t


3. d = u t + a (t^2) / 2

where:

• u and v are the unitial and vinal (sic) velocities respectively;


• t is the time;


• d is the distance travelled in time t; and


• a is the constant acceleration

For your case you know u, v and d, and need to know a, so use formula (1), rearranged into:
a = (v^2 - u^2) / (2 d).

(2) and (3) can then be used to calculate v and d at each time until arrival.