Chenkel said:
if I launch at an angle it will have a velocity tangential to the vertical velocity, and this vertical velocity will change due to the acceleration of gravity, but I don't see why this tangential velocity is changing direction to achieve a circular or elliptical motion, obviously I'm missing something and I'm trying to fill that gap in my knowledge, if anyone can provide that insight I will be grateful, thank you!
What you're missing is that the tangential velocity IS changing. Let's set up an example:
Let's place a stationary point-mass (a mass with zero size) at the origin of a graph. Let's also place a point-particle with substantially smaller mass at (0,1), one unit above the mass, and let us also say that this particle is moving in the +X direction at 1 unit per second.
Our particle is attracted towards the mass via a force. This force itself is a vector that points from the particle towards the mass. Initially this force cannot act on the particle's X-velocity since it is acting solely in the Y-direction. But, if we advance by a second (ignoring the Y-acceleration that would happen in that second under the applied force) we find that the particle is at (1,1) on the graph and now the force vector CAN act on the X-velocity since the force vector now acts equally in the X and Y-directions.
But wait, we just skipped a whole second when advancing time. Perhaps things are different in a continuous world. As a test, let's see what happens when we only advance a half-second. Well, the particle is at (1/2,1) on the graph, the force vector still has an X-component, and the particle's x-velocity will be changed, just by a smaller amount than when we advanced straight to 1 second.
If we continue to make our time steps smaller, we find that ANY deviation in the X-direction away from zero will result in the force vector having an X-component, and thus affecting the particle's X-velocity, which was the original tangential velocity. So when you throw a ball on Earth, it IS being pulled back towards you by the Earth's gravity. Just not by very much in that short time and distance.
But, what about the Y-velocity? Well, it turns out that something similar happens to it as well, it just starts at zero and increases instead of starting at one and decreasing. In a perfectly circular orbit both the X and Y velocity vectors 'trade' magnitudes in such a way as to ensure that the overall velocity vector has the same magnitude at all times.