This just isn't true. If you bored through the earth, assuming it was possible, which it isn't, the train would fall under its own power but as it came closer to the center of the earth the slower it would fall, when it reached the center of the earth gravity would hold it there. Everything would be up from the center and as the speed at which the train fell would gradually slow to almost nothing before it even got to the center, it would never get past the center without some type of power input.
Actually, if there was no air (or other) resistance in the tunnel, it would accelerate until it reached the center of the earth, and continue on its’ path, decelerating all the way back to the surface (or at least station level) on the other side.
However, zero resistance is a practical impossibility, but it WOULD rise from the center back up towards the surface.
The RATE of acceleration would drop as the vehicle went deeper, and would technically be a zero acceleration at the core, but the intrinsic velocity would remain, and thus deceleration would start as soon as it passed the center mark. . .
Gotta disagree with that one. The behavior would be like that of a pendulum. Given that practically there would always be some kind of friction, an additional boost would be needed to be sure the train got all the way to the other end. But no it wouldn’t stop in the middle any more than a pendulum does.
Actually, it would pass the center of the tunnel at maximum speed, then decelerate until it was almost to the far end (but not quite, because of frictional losses), and then if not stopped would begin to fall back toward the center.
Unaided, it would oscillate back and forth across the center of the tunnel, until all kinetic energy was dissipated due to friction, at which point it would (as you say) come to rest in the middle. But it might take a very long time.
With propulsion available, there would be no reason not to use a little of it during the trip so that it would make it all the way to the far end, where of course you would stop it.
For short tunnels, there would not be much pull from gravity, so you’d have to use mostly propulsion. But for long ones, you’d get a noticeable boost from gravity. How useful this amount would be, I have not bothered to calculate.
(Consider the angle of descent at the beginning of a loooong tunnel vs. a short one, assuming of course that they were straight-line through the earth.)
Of course this is all quite fanciful until they figure out how to make a thousands-of-miles-long vacuum chamber and transport people safely through it. This is 22nd century stuff, at best.