Today I was prompted (ha ha) by a question Anders Sandberg asked the latest GPT: at what angle should you throw a projectile — subject to uniform gravity, and ignoring air resistance — in order to maximise the length of its trajectory?
[The Chatbot got quite close, but fell over at the end, being too lazy to compute an easy derivative, and then screwing up its numerical solution to a needlessly difficult formulation of the problem.]
It’s well known that to maximise the horizontal distance travelled, you should launch the projectile at 45°. But to maximise the distance along its parabolic path before it hits the ground, the angle turns out to satisfy the beautiful transcendental equation:
s arctanh(s) = 1
where s = sin(θ)
This is solved by:
s ≈ 56.47°
Greg Egan /
npub12c…gd38p
2024-09-13 14:44:01
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2024-09-13 14:44:01Event JSON
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