The problem (long story short) to use an analogy... if I stand here, and then immediately run 5mph for an hour, after 30 minutes I have run 2.5 miles. But if I instead linearly drop from 5mph to 1mph over the course of that hour, where do I stand after 30 minutes? The speed is linear so I don't think this quite breaks into calculus, but I've been banging my head over this for a while. Reasoning: I'm trying to program a particle engine and calculate movement without storing origins since they won't be drawn on screen unless they're within the view frustum and therefore particles off screen that are moving around that suddenly come on screen would not be in the proper place. So the calculations have to be made without any state variables.