Why do objects follow different trajectories in freefall and orbit?

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Objects in freefall do not actually follow parabolic paths; they move in elliptical orbits around the Earth's center of mass, with the orbit sometimes intersecting the Earth's surface. When the dimensions of the orbit are vastly different, such as a minor axis of a few dozen yards and a major axis of 4000 miles, the elliptical path can appear nearly parabolic. A hypothetical scenario illustrates that if the Earth shrank to the size of a marble while retaining its mass, a thrown ball would still follow an elliptical orbit. While some argue that the trajectory of thrown objects can be approximated as parabolas near the Earth's surface, this is a simplification rather than an accurate representation. Ultimately, the discussion highlights the nuances in understanding object trajectories in relation to gravitational forces.
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Why do objects fall in parabolas, but things orbit in either ellipses or hyperbolas?
 
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Vorde said:
Why do objects fall in parabolas
They don't.

They move in an elliptical orbit about the centre of mass of the Earth; it just happens that the orbit intersects the surface of the Earth.

When the minor axis of an ellipse is on the order of few dozen yards, but the major axis is on the order of 4000 miles, an ellipse looks very much like a parabola i.e. the focus (of the ellipse) is so far away it might as well be at infinity (i.e. a parabola) for all the difference it makes.

If you tossed a ball in the air, and and that moment, the Earth suddenly collapsed to the size of a marble while keeping its mass, your ball would fall all the way to the centre, round it (with a perigee of just a few dozen yards) and come back out, following that elliptical orbit.
 
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So really whenever anyone tells us that things fall in parabolas, they are lying. Makes sense, but still...

Thank you for the answer.
 
Vorde said:
So really whenever anyone tells us that things fall in parabolas, they are lying. Makes sense, but still...

Thank you for the answer.

I don't think that is a fair statement. We do not have instrumentation that can separate the path of a thrown rock from a parabola. In fact, under the assumption that you are close to the Earth's surface, our basic models say that it IS a parabola.
 
I don't disagree, but I went through a whole Analytic Geometry class with the term 'things fall in parabolas' etched into my brain. I'm surprised this wasn't at least mentioned.
 
Vorde said:
I don't disagree, but I went through a whole Analytic Geometry class with the term 'things fall in parabolas' etched into my brain. I'm surprised this wasn't at least mentioned.

They are parabolas to an acceptable degree of accuracy. Two plumb bobs 10 yards apart, both pointing at the centre of the Earth, will be parallel to within one part in 700,000.

There are many factors affecting ballastic trajectories that are much greater than that.
 

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