Question

Extrapolate what you’ve learned about E-fields of a sphere above to the gravity-field of a massive sphere. If we were to make a tunnel through a diameter of the Earth we could create a nearly energy free way in which to transport items from one side of the Earth to the other. Calculate how long it would take for something to travel through such a tube to get from side of the earth to the other.

Answer 1

The circumference, or distance around the Earth, is approximately 40,075 km, but that depends on where you measure it; around the equator, or from pole to pole. So, to travel overland from one location to its antipode, you'd need to travel 20,037 km.

A tunnel, dug from one side of the Earth to the other would be, on average, 12,742 km. So it's a shorter trip, sure, but that's not the best part.

If you jumped into the tunnel, you'd fall down towards the center of the Earth, accelerating constantly, thanks to gravity. By the time you reached the halfway point, after falling for 21 minutes, you'd be traveling at 28,000 kilometers per hour.

Once you crossed the halfway point, the velocity would carry you back up the other side of the tunnel for another 21 minutes. This time, however, gravity is slowing you down, so by the time you reach the other end, you come to a perfect stop, just as you arrive at your destination. In other words the trip didn't require any energy. You exchanged gravitational potential energy for kinetic energy on the way down, and then exchanged it back on the way up again. No energy was created or destroyed. We obey all the laws of thermodynamics here on the Guide to Space.

The trick is that you need to make sure the tunnel is a complete vacuum, so that you don't experience any air resistance during your journey. That would cause you to fall at terminal velocity, and you'd end up stuck at the center of the Earth, completely weightless and helpless.