Driving into work, I started thinking about the arc of something being thrown and was puzzled about how gravity's affect is squared per second for falling bodies. Intuitively that implies the shape would be "like" a parabolic shape. But I'm curious if it truly is parabolic. Doing some googling, I've found some rudimentary explanations that remind me of explanations I received in algebra as a kid (such as http://entertainment.howstuffworks.com/physics-of-football1.htm)
My confusion is around the idea that gravity's effect on falling bodies (reference http://en.wikipedia.org/wiki/Equations_for_a_falling_body) is increased over time. If this is the case, then wouldn't this affect the arc in such a way that it's longer on the release end and curves downward faster at the end? This might not be as noticeable for short distances, but intuitively it could play an important role in longer distances.
In other words, my question is pretty simple, as soon as something is thrown (shot, projected, etc) is it considered a falling body. Either way, is the arc or path truly parabolic, or is the path elongated on the throwing end and curve downward faster at the end? If it is truly parabolic, can you please give a clean explanation as to why the effect of gravity over time doesn't apply? If my intuition is correct about it being elongated, can you please share a useful reference as well?
A couple of assumptions:
- What is being thrown is small and close to a large body. Like throwing a football 1,000 miles across to the earth's surface.
- Ignore air resistance or other factors for simplicity sake.
The Newton cannon illustration in @HariPrasad's answer shows us the flight path is elliptical not parabolic. It shows how modifying the initial vector's magnitude, when the angle is tangent to the earth effects the ellipse. It however does not show how changes to the initial vector's angle affects the ellipse.
Can we formulate an equation for the path (Reference: https://en.wikipedia.org/wiki/Ellipse)? I'm hoping for an answer that explains how the foci positions, and sum of red and blue line (need editor to give technical name) change in relation to changes in the direction and magnitude of the initial vector.