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I want to model a particle with an arbitrary initial velocity, and estimate the time it takes to reach a final point given a constant magnitude of acceleration. It should take the quickest path to the point. This takes place on a 2D plane.

I can't figure out how to model the acceleration for the particle, since when I think about the problem it seems like the acceleration direction should change as the particle's position relative to the final point changes. I don't know how it would change, or even why. Part of me thinks that the acceleration should be in a constant direction, but I can't figure out which direction that would be either.

As you can see, I am fairly lost here. Is there a general equation for the ideal direction of acceleration here? Thanks for any help you can provide. If it is at all relevant, this is for a simulation I'm writing in Python to predict the time it takes a particle with some initial velocity vector to reach a given point.

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    $\begingroup$ Sounds like a problem for optimal control theory: en.wikipedia.org/wiki/Optimal_control. You can start with a simple question: for the same position and velocity, does accelerating in any other direction than directly at your destination get you closer in an infinitesimal time $dt$? So if you keep accelerating towards the destination point in the optimal direction, is that not, at least, a locally optimal solution? $\endgroup$ – CuriousOne Apr 29 '16 at 18:14
  • $\begingroup$ It's like the "brachistochrone problem" solved with the Calculus of Variations but with main differences : Therein the acceleration vector $\mathbf{g}$ is constant (magnitude $g$ and vertical downwards) and the curve of "brachistos chronos" is a surface on which the particle moves without friction downwards under the influence of gravity. $\endgroup$ – Frobenius May 1 '16 at 10:39

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