Question:
My question is about orbital paths and projectile motion.
A diagram in my Astronomy text (Astronomy Today, Chaisson & McMillan, 4th
edition) shows the elliptical shape of all orbits, including those that
are "interrupted" by the Earth's surface. From this diagram, one would
conclude that projectile motion should take an elliptical path, contrary
to what we see in every physics text. Further, the diagram shows that
objects that achieve escape velocity have hyperbolic paths. Shouldn't a
parabolic curve be possible for speeds in between ellipses and
hyperbolas? Is the diagram wrong?

Replies:
The various conic sections taken by a projectile depends upon whether the
kinetic energy of the projectile is less than that accounted for by gravity
(a parabola), equal to i.e. in "free fall" (an ellipse), or greater than (a
hyperbola).

http://en.wikipedia.org/wiki/Parabolic_trajectory

gives a more quantitative algebraic treatment.
For those wanting even more details and are willing to navigate a large
physics web site try:

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