What if you fell into a tube through the earth?

Dear Cecil:

You may recall that Alice wondered about how deep the tube was while she was falling down the rabbit hole into Wonderland. If such a frictionless tube actually did exist, and went from the North Pole to the South Pole, and you or I accidentally stumbled into one of its open ends, exactly what would happen to us?

Cecil replies:

Whenever this job starts to get boring, I know I can count on one of you screwballs out there in Baltimore to come up with something. You guys are a national asset.

What we are dealing with here is essentially a cosmic pendulum. But let me explain a few basic concepts first. For one thing, we are going to disregard the problem of the earth’s molten interior. There is no surer way to ruin a good discussion than to contaminate it with the facts.

Next, you must force yourself to accept the following notion: if you were somehow teleported to a cave in the center of the earth, you would find that you were weightless. This is because you would have approximately equal amounts of mass on all sides of you, which would cancel each other out.

Now then. If you jumped into a frictionless (and consequently airless) interpolar tube, you’d fall, obviously, picking up momentum as you went. As you approached the center of the earth the pull of gravity would decline and eventually (at the center) cease, but inertia would keep you going.

Once past center, though, the pull of the earth’s mass behind you would begin to slow you down, at exactly the opposite rate that you’d accelerated. You’d come to a complete stop just at the brink of the Antarctic end of the tube, where you’d have an opportunity to wave gaily to the bunny rabbits or whatever they have out there before beginning to fall back in the opposite direciton. This process would continue forever.

Once we start figuring for the effects of atmospheric friction, of course, the situation changes. After a certain point in the course of falling you’d reach a top speed called “terminal velocity,” where air resistance would counteract the accelerating effects of gravity. With less momentum, you’d only fall a relatively short distance past the center of the earth before you stopped and started heading in the other direction. Eventually you’d reach equilibrium at the earth’s center.

I was going to calculate how long this would take, but twenty minutes of computation has produced no useful result and you didn’t ask anyway. But watch where you’re going.

Send questions to Cecil via cecil@straightdope.com.

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