Parabolic paths vs Elliptical paths.

[mprm86]

We are always taught that a projectile describes a parabolic path (neglecting air resistance), but the path is actually elliptical. So, my question is this: A projectile is thrown in point A (on the ground), it reaches a maximum height H, and it finally falls in point B (same height as A, that is, the ground). Which will be the difference between the paths if (a) it is elliptical, and (b) it is parabolic? Any ideas, suggestions?
Thanks in advance.

P.S. The answer I´m looking for is one of the kind of 1 part in a million or somewhat.

 

[mathman]

We are always taught that a projectile describes a parabolic path (neglecting air resistance), but the path is actually elliptical.

Where did you get this idea? The path is parabolic. You can approximate a parabola by an ellipse as close as you want by simply moving the foci farther apart. The parabola can be looked at as the limit as the separation becomes infinite.

Added note: You may have a point since the Earth is not flat. The distant focus will be the center of the earth.

 

[rcgldr]

The classic parabolic path assumes a flat earth.

If the projectile travels below escape velocity, the path is elliptical.

If the projectile travels exactly at escape velocity, the path is parabolic.

If the projectile travels faster than escape velocity, the path is hyperbolic.

A link for some formulas (go to orbital mechanics page)

http://www.braeunig.us/space

 

[Galileo]

What's responsible for an elliptic path (if v< v_escape) is not the curvature of the earth, but the variation of the gravitational force with height.
You could solve Newton's law under a inverse square force field to find the actual path. The variation g with height is very small to take into consideration when throwing stuff in the air though. (Air resistance is WAY more dominant)

 

[rcgldr]

What's responsible for an elliptic path (if v< v_escape) is not the curvature of the earth, but the variation of the gravitational force with height.

No one mentioned curvature of the Earth in this thread. My reference to a parabola being correct for flat Earth was a reference to treating gravity as being effectively generated from a flat plane instead of effectively from a point source (in which case you get an elliptical path).