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  • $\begingroup$ Thank you. However that does not explain how to get the maximum exit speed and best angle from a curve that descends from height h and ends at height zero. I want to know the curve that will produce the maximum range. This does not necessarily end at 45 degrees. For example, if the exit speed at the end of the track is greater following the descent along a particular path, then this extra speed may compensate for the less than ideal angle. On the other hand, a track that is constrained to end at 45 degrees may not produce the greatest exit velocity. This is not a simple problem. $\endgroup$
    – chasly - supports Monica
    Commented Mar 25, 2023 at 19:34
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    $\begingroup$ @chasly-supportsMonica, as the answer says, the curve doesn't matter. All curves from A to B that are smooth enough to not cause drag leave the ball with the same energy at B. Only the exit angle matters. $\endgroup$
    – BowlOfRed
    Commented Mar 25, 2023 at 19:34
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    $\begingroup$ @chasly-supportsMonica The exit speed (not velocity) is independent of the path, as required by conservation of energy. Yes, this is the claim. $\endgroup$
    – Puk
    Commented Mar 25, 2023 at 19:37
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    $\begingroup$ The Brachistochrone is special in that the ball reaches B in the earliest possible time. But it has the same speed at B as other curves. $\endgroup$
    – BowlOfRed
    Commented Mar 25, 2023 at 19:38
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    $\begingroup$ @chasly-supportsMonica Sure, I assumed that was fixed since it's specified in your diagram. $\endgroup$
    – Puk
    Commented Mar 25, 2023 at 20:02