A list of ten things that you didn't know about the earth. My favorite one, by far:
But what if you did dig a hole through the Earth and jump in? What would happen?
Well, you'd die (see below). But if you had some magic material coating the walls of your 13,000 km deep well, you'd have quite a trip. You'd accelerate all the way down to the center, taking about 20 minutes to get there. Then, when you passed the center, you'd start falling up for another 20 minutes, slowing the whole way. You'd just reach the surface, then you'd fall again. Assuming you evacuated the air and compensated for Coriolis forces, you'd repeat the trip over and over again, much to your enjoyment and/or terror. Actually, this would go on forever, with you bouncing up and down. I hope you remember to pack a lunch.
Note that as you fell, you accelerate all the way down, but the acceleration itself would decrease as you fell: there is less mass between you and the center of the Earth as you head down, so the acceleration due to gravity decreases as you approach the center. However, the speed with which you pass the center is considerable: about 7.7 km/sec (5 miles/second).
Fast forward to the year 2483 and we'll probably all be using such holes to quickly travel through the earth. Spain to New Zealand in 42 minutes! New York to the middle of the Indian Ocean? 42 minutes! I also recall reading somewhere that the tunnels don't need to run through the middle of the earth. You don't get the free fall effect, but with the proper contraption (mag-lev tunnel train?) you'll be pulled through the tunnel at a great speed. Does this ring anyone's bell?
Update: A bell has rung. The tunnels described above are called chord tunnels and the travel time through the earth in a frictionless chord tunnel is always 42 minutes, even if the tunnel is only a few hundred miles long or so (say from New York to Detroit). (thx, mike)
Update: In this short Nova clip, Neil deGrasse Tyson "demonstrates" a trip through the center of the earth. (thx, michael)