There are a few points where I feel the author might have been taking force/velocity diagrams a bit too literally (and is a bit sloppy with the difference between force and velocity).
> the following main velocity vectors acting on it:
> gravity acceleration
> thrust
> rocket's velocity
and
> Once in orbit, the spacecraft will have two main forces exerting their grab onto it: the tangential velocity and the gravitational pull.
I'm not sure whether someone new to those concepts would notice, but it's a potential source of confusion.
> Gravity Turn or Pitchover is the second maneuver that is executed as early as possible by using the gimbal of the engines or by using cold gas thrusters on the nose of the rocket or a combination of the both.
Honest question here: do any rockets use their cold gas/RCS thrusters for the initial pitchover? I feel like they wouldn't be powerful enough in most cases to pull it off.
> The velocity vectors are similar to the vertical flight phase but because the gravity acts on the same vertical plane it makes the spacecraft change it's pitch without additional input from the engines, tasking them with the only job of increasing the speed of the spacecraft.
Technically, it's a combination of gravity and aerodynamic forces that cause the pitch to change. Gravity causes the velocity vector to turn, but doesn't exert a (noticeable) torque. It's aerodynamic forces that work on the rocket to (hopefully) keep it aligned with the velocity vector.
This is why some rockets have fins at their base - additional drag at the base ensures that the aerodynamic forces keep the rocket pointed the right way. Rockets without enough drag near their base or too much drag at their nose will tend to flip right around with interesting consequences.
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That being said, it's one thing to read about orbital mechanics, but it's hard to beat hands-on experience for really wrapping your head around things.
As mentioned in the article, Kerbal Space Program is one frequently-recommended way to go about this, and does a fairly good job with the basics. Not to say that the basics aren't much; you can get quite far with "just the basics", and arguably they'll work just fine for the most common mission profiles.
KSP does use a simplified gravitational model, though, taking into account the gravitational forces of only a single body at a time, which means it's missing some more interesting features of full n-body dynamics, such as Lagrange points and low-energy transfers. If you're interested in those, consider trying the Principia mod, which adds n-body dynamics, non-uniform gravitational fields, and more [0]. There's also the Realism Overhaul mod if you wish to work with more realistic rockets and celestial bodies [1].
[0]: https://github.com/mockingbirdnest/Principia
[1]: https://forum.kerbalspaceprogram.com/index.php?/topic/155700...