I had a strange nightmare last night... It relates to this problem.
I was in my car rolling down a hill, thinking about this problem (don’t know why the engine wasn’t running). I guess I wasn’t paying attention to the road, and ran into someone

(I think it was my physics teacher...). I almost woke up. Almost, but not quite.
The collision sent him over the edge of the road, perpendicular to my path, at the same velocity as I was traveling at the time, with only a horizontal component. I know his velocity was the same as mine, as I could see him in my side mirror, and he was always at an angle of 45 degrees on my left.
As he went over the edge, his path became parabolic in the z direction, just as a roller coaster would actually be built to provide that freefall feeling. He was neglecting air resistance (wouldn’t you at such a time?).
I continued down the slope, and followed it to the left, going up and down several times, at varying angles, finally becoming parallel to the path of the teacher, at the bottom of the hill.
Now he was on a roller coaster (told you this was a strange dream), and he was out ahead of me (glad he is OK, I was worried).
What I saw woke me up.
He was out ahead of me because he reached the bottom before I did. We had the same velocity, he was not getting further away from me, neither was I catching up to him.
So, my intuition tells me that time is in some way proportional to the angle of descent, but only the starting altitude affects final velocity. More importantly, final velocity is completely independent of time and angle of descent.
The physics tools I have at this time are kinematics, circular motion, and momentum. Can I prove this conjecture? If I can, then the velocity at the bottom of the track is the same as if the coaster was simply dropped off the edge of the track (of course I mean magnitude of velocity, not direction).
Regards,
Jim