You get to orbit by going fast enough, so that acceleration due to gravity acts perpendicular to your velocity and so acts to change your direction and pulls you around the earth.
Imagine you are standing on Mt Everest (and the atmosphere has magically disappeared), and you would like to shoot a bullet once all the way around the earth, how would you have to fire it? It seems pretty intuitive that shooting it up into the air won't do the trick. If you want to shoot around the earth, you need to fire in parallel to the earths surface, with a sufficiently high muzzle velocity. And really an orbit is nothing else than a shot around the earth.
I did not do the maths.
You only turn ever so slightly heavenward, mainly to avoid air resistance.
> When you start tangential, as soon as you cover a distance equivalent to the radius of the earth, you are then perpendicular (maybe not directly so but there is no practical difference as far as the gravity well is concerned).
If you were assuming the tangential take-off would continue going a straight line instead of curving in an orbit, I still don't get what you mean -- after covering one Earth radius, you'd be traveling at a 45-degree inclination relative to the surface of the planet, not perpendicularly.
That's not what being in orbit is. In fact that's the opposite of being in orbit. To be in orbit you need to move parallel (tangent) to the surface of the earth, not perpendicular.
The distance about the surface is entirely for air resistance, and has nothing to do with being in orbit.
> ... accelerate upward then turn 90 degrees and accelerate horizontally
I am not for such a scenario at all. The point of the upward (upward only and not considering the atmosphere) acceleration to the desired location is to give the lower bound of the energy requirements. This is the baseline (baseline-1) and the rocket equation is as simple as possible.
In reality with an atmosphere and to put the object in orbit, the aerodynamics change and the energy requirement increases beyond the above baseline-1.
If you "accelerate upward then turn 90 degrees and accelerate horizontally", you can calculate an energy requirement for that and it is easy. Only two vectors involved. That should give some limit (call it baseline-2). We should expect to do better than baseline-2, how better? A calculation using the diagonal of the vectors involved in baseline-2 should give us baseline-3.
We shouldn't do better than baseline-3. Our launch designs and ingenuity should have an energy requirement between baseline-2 (this is bad, we are not thinking) and baseline-3 (this is maybe closer to ideal).
The rockets and shuttles do "pitch-over manoeuvres" to turn the straight upward acceleration into an elliptical acceleration.
* Note, I have not addressed the complications of the variations in atmospheric drag, but if it varies close to linearly along the vertical cross-sectional then how I think about it above does not change unless there is some other oversight.
Doing that with a track would be expensive because the track would have to be built hundreds of miles high over all of its length. It would be cheaper to build most of it lower, and maybe accept that we'll have to handle the air resistance somehow. If we build a track that doesn't go out of the atmosphere, we could still use it to build up a lot of speed and then turn the rocket upwards before the thing is self-powered. If we do build a track that goes out of the atmosphere, we'd still want to get as much ground-level acceleration as we can.
Maybe it's more practical to build the track on the Moon, where there's no atmosphere.
What I was grappling with, is. I presumed @ars (parent) knew he was talking and in expressing my contention it would get addressed with a little bit more information in what I was missing. I now see some emphasis in his explanation and additional links.
The key is was that tangential acceleration opens up none fuel based acceleration mechanisms i.e change to the type of energy and the quantity (you accelerate less fuel to burn up the fuel).
(well, any combination of speed, direction, and position will be an 'orbit' in some sense, in that there's a conic section you're on that you would follow if you were in freefall. But if you're sitting still over a planet then it's the degenerate ellipse that's a straight line down to the planet's core).
So in the absence of atmosphere the ideal ascent trajectory would look pretty much like a Hohmann transfer orbit: you'd accelerate horizontally until you were in orbit at surface level, and then you'd do the minimum energy transfer from that orbit to a higher orbit. In reality it's worth getting to altitude where the air is thinner before turning horizontal, but even so, the vast majority of a rocket's acceleration is horizontal, not vertical.
You can do the maths, but if you want to really understand these things, play Kerbal Space Program. Seriously.
Or perhaps you are having a bad problem and will not go to space today.
And then you turn Earthward, to avoid crashing into the Earth.