Gravity elevators are an old idea in science fiction and consist of an evacuated tunnel dug straight through the centre of the Earth as shown below. The idea is that the elevator falls due to gravity until it reaches the other end.
Below ground level your acceleration due to gravity is not constant; it is proportional to your distance from the centre of the Earth. In terms of the coordinate system shown above, this relationship can be written; a = - g x /Re where g = 9.8 ms-2 is the acceleration due to gravity at the surface, and Re = 6378 km is the radius of the Earth. If an elevator falls from rest at the one end of the tube;
a) Calculate how fast it will be going when it reaches the centre of the Earth.
b) How fast will it be going when it reaches the other end?
c) How long will the journey take?