The 'Gravity Sled' - also referred to as the 'Gravity Train' or 'Gravity Express' - is a simple concept to describe, difficult to survive, and almost impossible to create; but it does offer the opportunity of really cheap travel or shipping at tremendous speeds.
Drill, dig, or otherwise excavate a tunnel straight down through the centre of the planet (any planet, but this one is the closest) and straight through to the other side. Line it so that the magma and its heat do not create an unhealthy environment, and perch a sled at one end. It doesn't matter which end, as both sides are 'down', and both act as a combination entrance/exit point.
Once that has been accomplished it's time for the fun. When something - anything that has mass and is heavier than air - is dropped down one of the ends of the tunnel, it will quite naturally fall toward the centre of the planet, accelerating all the way. After it passes the centre point, the same gravitational force will begin to cause the acceleration to slow at the same frantic rate that it had been increasing.
By the time the occupied sled reaches the other end, it will have lost all momentum and be ready for another go at the centre. There should be a mechanism designed to keep this from happening to give the occupants time to disembark. And all this takes place in just 42 minutes1, no matter what size the payload is!
It is worth noting that this theory is now over 400 years old. Gravity however, is about ten thousand million years old - possibly much, much more.
English physicist Robert Hooke, a contemporary of Sir Isaac Newton, who did amazing experiments with springs, may have been the first to see this possibility. He was a professor at Gresham College; and, as a brilliant mathematician and prodigious inventor, he made significant contributions to the sciences of physics, astronomy, and biology.
In his correspondences with Newton, Hooke explained how a body would move inside a planet, if there was no resistance. In the 19th Century, the Paris Academy of Sciences received (but later deferred on) a seriously-proffered gravity train project. In the next century, the gravity train also became a part of elementary mechanics textbooks.
US physicist Paul Cooper published a paper proposing the concept for a future transportation project - in the American Journal of Physics (Time Magazine, 11 February, 1966, p.42). Heated debates, though not as heated as the sled would get, followed.
Because some friction is inevitable, the sled should be dropped from a point higher above the earth's surface at its entrance point than the height 'catcher' is at the exit point, or have a slight bit of propulsion. This will help it to get all the way to that other end. Without the added momentum, friction would cause the sled to oscillate towards one end and back toward the other, losing momentum with each trip, until it ended up stuck at the centre of the planet.
The calculations show that, given no air resistance, the speed of the sled, travelling through the centre of the planet, would reach 11,170 metres per second (40,212kph), or around 25,000 miles per hour. Oddly, or perhaps necessarily, the speed corresponds well with our established planetary escape velocity.
To work, the tunnel would have to be evacuated of air, and the passenger compartment heat-proofed and self contained.
Even if it will never be possible for humans to make the journey, imagine how cheap transporting non-organic materials would be if you could just drop them to the other side of the world instead of having to fly, float or drive them around!
You CAN Get There From Here
The beauty of the gravity sled is that the sled's tunnel needn't pass through the centre of the planet to work! The really neat part is that any two cities that could create a straight-line tunnel between them - a tunnel that cuts across any arc of the planet's surface - could enjoy the same sled ride; albeit at less insane velocities and temperatures. And every tunnel's ride would take the same 42 minutes!
For trains travelling a lesser distance, perhaps one-sixth the outside surface of the planet, the speeds would be vastly lower. The beginning of the tunnel is still on the outside of the planet, being pulled in toward the centre, and thus would start its slow but accelerating trip toward the centre of the tube. As it passed the midpoint, it would ride back up towards the surface and would slow at the same rate, coasting uphill until it arrived at the exit after 42 minutes.
That's it. That's all there is to it. That's how gravity works it all out. It's always the same 42 minutes between any two points.