This physics problem, determining the launch angle that gives the maximum range when the starting point is above the level ground, is a personal favorite since it took me a year of cogitating to come up with this solution. No, I never threw anything off a building, but I heard about some rowdies who did, and that is what got me to thinking about this problem.

Of course, ignoring air resistance and other complicating factors means this is just an approximation (and quite possibly a bad one), but accuracy not the point. Such simplifications are useful for teaching the basic concepts of physics and geometry, and for the simple pleasures of exploring the mathematics involved.

BTW, please do not throw things off of buildings! Also, if you throw anything into a body of water, make sure it's natural, not trash.

The method I came up with to solve it involves no calculus or quadratic equations. All you need is elementary physics (conservation of energy, plus basic velocity vector concepts) and some elementary geometry (area of a triangle, simple trig functions). I can't be the only person to come up this solution, but I have have never found anyone who has seen it before. All the websites and videos about this problem use calculus and complicated trig identities. And none of them point out that the denominator in the answer is the final velocity of the projectile. If anyone has heard of this solution, please let me know in a comment.

Update: Commenters have pointed out that this solution is well known by Russian Physics Olympiad competitors.

Animations were created using Desmos Graphing Calculator ([ Ссылка ]) with the help of DesmosPlayer, a browser plug-in that I wrote to control the graph ([ Ссылка ]) Screen capture turned that into a video; edited with iMovie.

Music: “Off Broadway”, iMovie Song ([ Ссылка ])

Corrections:

1:35 “Ray” is the wrong term here. Velocity is represented by a “vector”. A ray doesn’t have a specific length. It extends infinitely in one direction.

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