Puzzle Description:

I liked the physics question posed by the earlier puzzle, so I'm posing my own here (with some obvious suspension of reality required): A) A person standing on the north pole digs a hole straight through the center of Earth. B) A satellite is in low Earth orbit, orbiting longitudinally (from pole to pole). Time is paused with the satellite (in orbital velocity) directly above the person standing beside the hole. Action starts and the person falls into the hole... Which ends up on the other side of Earth first, at the same distance from Earth's center as when the action began?

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

I don't see how the person in the hole makes it past the center of the earth (the center of earth's gravity), to fall "up" the rest of the way to the south pole

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.

I SO do NOT want to be the person yo-yo-ing through the center of the earth in this scenario.

(Not reading the other comments yet...)

One presumption is needed for this question: that the earth's density is consistent. Which is not the case, of course, but set that aside and just presume it is.

The person's falling is caused by gravity. So they will fall faster as they approach the center. But what happens then?

Now reading the other comments: And we all start by considering this same issue! Thank you, perlwolf, for the explanation.

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

assuming this is meant to be a simple race to the other side, and assuming that the Earth is not the source of gravity but is merely a tube for the person to fall though, we are merely comparing speeds. The person falling though the Earth pulled by an equivalent Earth gravity is still limited to Terminal velocity within our atmosphere which is approximately 120mph. The earth is approximately 8,000 miles in diameter, so it would take roughly 66hours and 40 minutes to fall through that distance.

A Low Earth Orbit (LEO) has an orbital period of 128 minutes. So in the time it take a person to fall that distance the satellite will have traveled the orbital distance 62.5 times.

In short, the satellite wins not surprising since the satellite is traveling nearly 142 times faster than the human.

Making the usual Physics assumptions that (1) the satellite just skims the surface of the earth without hitting anything, and that (2) there is no air resistance, then the person and the satellite start at the same distance from the center of the earth, and they will run into each other at the South Pole.

i.e., same time for a half-orbit.

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As to the physics question, I think the arrows are backwards. If they start at north pole and dig their hole straight through, then they finish digging at the south pole, so it's most likely that's the starting point of their fall right? I guess they can travel back to the north pole to restart from the north?

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.

Great discussion! So many truths in there, especially David's #9 pointing out my logic error. :) I propose therefore that the person is both at the South Pole upon completion of their digging -and- at the North Pole, due to Quantum entanglement, of course.

Sorry for the high difficulty of the solve; it was unintended. :) I rely on the [Check] button and if something doesn't work, I change something until it does.

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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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