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#1: besmirched tea (Besmirched Tea) on Oct 3, 2021 [SPOILER]

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IBT, it would depend upon whether there was air or vacuum in the hole. With no air resistance, the person would accelerate all of the way to the center, and decelerate all of the way from the center to the south pole, ending up at zero velocity whenreaching the pole (assuming that both poles are the same distance from the center of gravity of the earth and ignoring micro-gravity distortions from density fluctuations in the earth's composition. When you consider an ideal vacuum tunnel between any two points on earth, with the best "shortest-distance" parabola possible between those two points, it works out that all trips take exactly the same time - point pairs far apart get up to a higher speed because of the longer acceleration-deceleration time, closer point pairs have a shallower curve and don't accelerate as much (unless you add some sort of motor to help, of course).

With air resistance, especially with very high density air near the center of the tunnel, the falling person would overshoot the center a bit and then fall back and eventually stop at the center.

The falling person (back to going through a vacuum) is in a highly elliptical orbit (the limit of an ellipse is a straight line), with a smaller radius than any near earth orbit satellite than is not at ground level, so the satellite has a longer orbital period than the falling person. Reaching the south pole is actually a half orbit.

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No idea. Pretty image. Tough puzzle with lots of guessing at the end.

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First we would have to overcome the enormous engineering problems such as molten mantle and outer core and solid iron spinning inner core. Then we would cap both ends and pump out the air so that the falling object would not reach terminal velocity. This then becomes a calculus problem as the speed increases to maximum and the gravitational acceleration decreases to zero at the center then reverses on the way back out.

My calculus is as rusty as my memory but I did this calculation in my youth and I recall that it was about a 45 minute trip.

A satellite in low earth orbit takes about 90 minutes for a complete orbit so a half trip should be about...drumroll please...45 minutes. I think it would be pretty close to a tie.

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Found to be solvable with moderate lookahead by valerie.

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Some tricky (but manageable) logic, for an interesting solution and interesting discussion in the comments.

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