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A UFO is at one kilometer altitude, and one kilometer due east of a power plant on the ground it wants to investigate.

The UFO has infinite acceleration but is limited to a speed of ( altitude + 1km ) / 100 / sec. (Example: at ground level it can travel 10m/s. At 1km altitude it can travel 20m/s.)

What flight plan will get the UFO to his destination the soonest possible?

(You can ignore curvature of the earth, variance in terrain, and so on. Put the UFO at (1,1) on a graph and fly him to (0,0) if you want.)

Once you solve this I have two further versions of this problem at UFO in a hurry #2: beam up a car's driver and UFO in a hurry #1b: crash at a power plant

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  • $\begingroup$ Is this a logical puzzle or a calculus problem? $\endgroup$ – Vijay Apr 14 '20 at 10:07
  • $\begingroup$ it is a math problem $\endgroup$ – Swiss Frank Apr 14 '20 at 10:25
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Solution:

Consider the point $X$ 1 km due east of the UFO, and 1 km below ground. The optimal flight plan to the power plant maintains constant distance to this point.

Optimal flight path

Reasoning:

To see that this is indeed the case, place a Poincaré half-plane model of hyperbolic space onto the scene in such a way that the half-plane ends 1km below ground.

If the UFO flies at maximum velocity in euclidean space, its corresponding trajectory in hyperbolic space will maintain constant velocity. So the optimal flight plan corresponds to a shortest path in hyperbolic space, which is given by a geodesic, and in the Poincaré half-plane model the geodesic looks like a circular arc centered on the bottom edge of the model. The actual center of the arc is at Point $X$, since Point $X$ is the only point on the bottom edge that has equal distance to the UFO's initial position and the power plant.

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  • $\begingroup$ Is there a more intuitive explanation? $\endgroup$ – rinspy Apr 15 '20 at 11:32
  • $\begingroup$ This is by far more intuitive than any explanation involving calculus. But if you like, you can show that if you project any flight plan radially onto the circle around X, it will stay below the speed limit, so an optimal flight plan will always be found on that circle. $\endgroup$ – Magma Apr 15 '20 at 12:32

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