Fastest Path with Acceleration at Points

Question

This is just something I came up with on my own, but it seems like a fun problem and it has me stumped.

You have a set of points in two-dimensional space, with one point

Answer 1

You can try solving this problem backwards by recursively tracing paths from the end to each other node, then designate maximum speed along the line to be able to turn from that node to any other. The culling rule will be if a path from current to next node exists with less velocity and less time spent from end, which will mean that the other path is more optimal by default because it can reach more nodes and takes less time. Once a path reaches start node, it should get recalculated based on the maximum speed achievable at the start and stored. Then you gather the path with less time spent.

You have to search for any available path here, because the available paths on your graph are dependent on past state with an indirect mechanics, using less speed allows more choices.