how maximum range could occupy elliptical path in projectile motion i cant imagine how this happens if earth is sphere and the path is parabolic so still it wont make any of the elliptical path. then how could one say that its max range is ellipse?

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    $\begingroup$ I don't understand what exactly you are asking, but perhaps it helps to realize that the parabolic path for a projectile is obtained for a flat infinite earth (gravity always pointing in the same direction). $\endgroup$ – user1583209 May 12 '17 at 12:07
  • $\begingroup$ The path is elliptical not parabolic. Well, the path is parabolic if and only if the projectile has exactly escape velocity. It's elliptical if the velocity is below escape velocity and hyperbolic if the velocity is above escape velocity. Note that we frequently approximate the path as parabolic if the maximum projectile height is small enough that gravity can be considered constant. When the projectile height is great enough that the decrease of gravity with altitude becomes significant the approximation of a parabolic trajectory breaks down. $\endgroup$ – John Rennie May 12 '17 at 12:08
  • $\begingroup$ This is a reasonable question, if a little confused. Here is a picture of what John Rennie is talking about. en.wikipedia.org/wiki/Parabolic_trajectory $\endgroup$ – mmesser314 May 12 '17 at 13:22

We say the trajectory of an object is parabolic only because of the approximation of a flat earth, with gravity uniformly pointing in one direction ("down").

However, like you suggest, if we instead treat the earth as a sphere, with gravity pointing toward the center, the path will follow an ellipse.

Note that if the velocity of the object is exactly equal to the escape velocity of Earth, the path actually will be parabolic. If the velocity exceeds the escape velocity, the path will be hyperbolic. Hyperbolas, parabolas, and ellipses (and circles) are all "conic sections", meaning they take the form of cross-sections of cones. In the end, they are all very related to each other.

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