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Falling Through the Earth

Date: 05/27/2002 at 11:05:47
From: Josh
Subject: Earth Question

What if someone dug a hole straight down all the way through the 
earth and then jumped into it.  What would happen when he came out 
the other side?  Would he blast off into the sky or would he fall 
back down into the hole?


Date: 05/27/2002 at 15:42:02
From: Doctor Tom
Subject: Re: Earth Question

Hi Josh,

It depends on where the hole is!

Let's start with an artificial situation -- the earth is
not spinning, and there is no atmosphere which would cause
friction on the falling body.  Then if you jumped in, you
would fall faster and faster until you reached the center,
at which point your speed would be greatest.  As you
continuted "up" the hole on the other side, you would slow
and slow, and come to a stop exactly at the surface at the
opposite end of the earth.  If you didn't grab the edge
and pull yourself out, you would then repeat the motion
exactly, dropping to the center and returning to your
same place.  Without grabbing edges or friction, you
would cycle like this forever.

If there's friction, you would still speed up until the
center, but more slowly, so when you got to the center,
you wouldn't be moving so fast.  Then you would rise up
the opposite side, but not all the way to the top, where
you'd stop and start falling again.  Due to more friction,
you would continue to lose speed on each cycle and would
go back and forth from the center, each time stopping
deeper in the earth.  Eventually, you would stop in the
center.  The more atmosphere (and hence friction), the
more quickly you would slow down.

Now, with the actual case of the spinning earth, if there
were no friction from atmosphere, you would cycle together
if the hole were drilled from pole to pole, but from
any other point, you would slam into one of the walls on
the way down.  To see why, imagine, that the hole goes
from somewhere on the equator to its opposite point,
also on the equator.  At the surface of the earth, you
are moving with it at about 1000 miles per hour.  Without
friction, you'll continue to move that fast, but half
way down, the earth is only turning at 500 mph, so (if you
got that far) you would be moving sideways in the hole
at 500 mph (relative to the speed the hole was moving).
So unless it was a very wide hole, you'd hit the side
soon after jumping into it.

With the actual atmospheric friction, you'd hit the wall,
and your speed would be less than before.  With atmosphere
and jumping north pole to south pole it would be just
like I described above for a non-spinning earth, but
with atmospheric friction.


- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Physics/Chemistry

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