# How Can I Get This Equation From These 3 Equations

1. Oct 13, 2013

Last edited: Oct 13, 2013
2. Oct 13, 2013

### fzero

I don't believe that the coordinates $x$ and $y$ defined here will satisfy an elliptical equation. These are the equations for the position of an object propelled into a ballistic trajectory on Earth. If you solve eq. 2 for $t$ and then substitute into eq. 3, you will get the equation of a parabola:

$$y= - \frac{g}{2v^2 \cos^2\theta} \left( x - \frac{ v^2 \sin(2\theta)}{2 g } \right)^2 + \frac{ v^2 \sin^2\theta}{2 g }.$$

This is an approximation to the true elliptical gravitational orbit of the object. The reason we don't get an ellipse is because $x$ is a flat-Earth coordinate that does not follow the direction of the true curvature of the Earth's surface. In polar coordinates the elliptical nature of the orbit should emerge. I believe that this article should have a correct analysis of the situation.

3. Oct 14, 2013