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End position of movement factoring in deceleration

Ok I am hoping to apply this answer to a piece of software, but it uses physics to work out the result so I require some help in that department. I will attempt to explain this the best I can. I ...
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Numerical simulation of mechanics problem

Say I have a planet and shoot something with a given velocity, which is a significant portion of the escape velocity, in a given angle into the sky. It has some initial velocity and there is the force ...