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I think this is more difficult than it appears.
I'm fairly sure you don't need to go faster than about 184 m/s or 665 km/h.
According to the formula above, that's the speed of the day/night terminator (i.e. sunset) on the Arctic or Antarctic Circles, which by definition are the furthest latitudes through which the terminator passes every day.[Edit: sunset is not instantaneous / the terminator isn't a sharp line, so you can probably trace a circle about 5 degrees inside your polar circle, and maybe go about 20% slower than that].
But is there a more optimal path..?
If I want to ride off into the sunset on a horse, how fast do I need to ride for that sunset to never end?