The problem is: Consider a tunnel that connects any two points A and B on the spherical Earth (assuming constant density). The tunnel is a vacuum, and the train traveling through the tunnel is travelling on a frictionless track with no engine (i.e. it is just falling through the tunnel). Show that the transit time is independent of the positions (e.g. San Fran to LA, or NYC to Sydney). How long is the ride?
The Attempt at a Solution
The only thing that I can think of is integrating the function that determines the g value to get the velocity and somehow showing that has a constant time not dependent on the distance that the "train" travels. Could anyone give me some insight? Thanks in advance.