So of course both of these are slight simplifications, but what is the connection between the two? If the earth is basically a circle, is an ellipse just a parabola stretched around a circle? Is a parabola just an approximation of a tiny part of an ellipse? How high do you have to be before you change your calculations of a trajectory?
The Math ain’t mathing.
Geometry!
https://en.m.wikipedia.org/wiki/Conic_section
They are similar and related but not quite the same thing. It does come down to whether it is orbiting or returning to a ‘flat’ plane, and while an eclipse isn’t quite the same thing as two parabolas put together, that isn’t too far off.
If something is moving in an elliptical orbit, and you are observing it from a fixed point in space, it will appear elliptical.
If something is moving in an elliptical orbit, and you are observing it from a body that is itself also moving in an elliptical orbit, it will appear parabolic.
Lots of answers touched the correct answer, which is that in reality things don’t follow parabolas on earth, a parabola is just close enough to the actual thing the object is doing to be indistinguishable. In reality everything follows elliptical orbits, but the top of an ellipsis with a Major axis of 6378 km and a few meters in the minor axis looks the same as a parabola, especially when you don’t see the full orbit because the object hits the ground. If you were to throw a rock and suddenly the entire earth besides that rock collapsed to a single point, your rock will orbit earth in an elliptical orbit.
It’s a parabola because the rest of the ellipse is in the ground.
Aren’t ellipses basically two joined halves of parabolas?
You might like to try Kerbal. You can see a parabola become an ellipse as your velocity and altitude reach a point where you miss hitting the planet on the way back “down.”
I think that mental model only works if you imagine the parabolas as reaching to infinity in a finite space so that both ends are parallel, ie having identical vertical slopes of +/- infinity. At that point, easier just to call it “half an ellipse”. To me, it’s much easier to imagine a parabola as the end of an infinitely long ellipse.
Your intuition and the KSP example are correct though. If you imagine the plane and cone for a parabola, you wouldn’t notice any significant change to the shape (at a finite distance) if you tipped the plane ever so slightly into forming an ellipse (or a hyperbola, for that matter) since it’s all smooth changes.
Anyway, the size of the elliptical (I think hyperbolic would have a different sort of energy state) arc that’d be formed by a thrown object would be so large relative to human scale as to basically be infinite, equivalent to a parabola. I imagine the difference might become significant once you are launching something a decent way around the Earth, but with that much energy in play I don’t think it makes much difference where exactly the projectile “lands”.
Wouldn’t things only be as infinite as your zoom level? Zoom out to solar system, zoom out to galaxy, etc?
They move in ellipses on earth as well, but it’s cut short by hitting the ground. Also, air resistance affects the trajectory so that anything thrown/shot will eventually return.
Source: Thousands of hours in KSP
Only stable orbits are elliptical. Many objects follow parabolic and hyperbolic trajectories in space. Eventually, many intersect with other objects. As time passes, objects that are not in stable (or semi-stable) orbits become less common through attrition.