How do speed and position relate to one another?

If one knows the POSITION and SPEED of an object at a given time, one can then predict the position of that object relative to the original POSITION at any time afterwards. Also useful to know would be the DIRECTION in which it was moving - which would enable a more accurate prediction of it's subsequent position; and any rate of ACCELERATION (or its negative, deceleration), i.e. the rate of change of speed, as this would also affect the object's subsequent position.