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How Can I Get This Equation From These 3 Equations

  1. Oct 13, 2013 #1
    Last edited: Oct 13, 2013
  2. jcsd
  3. Oct 13, 2013 #2


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    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.
  4. Oct 14, 2013 #3
    Thank you. But my teacher said he had proved the equation. But he didn't prove it t for me
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