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Understanding Newton's Mountain

The cannon is atop an imaginary mountain far above the atmosphere.

(The backdrop diagram is from Sir Isaac's own book, A Treatise of the System of the World, page 5.)

For low speeds, the cannonball doesn't get far relative to the Earth's size, and the downward gravity gives a parabolic path. At higher speeds, the cannonball goes far enough that the direction of gravity--always toward the center of the Earth--changes, and so does the shape of the path!

Link to lecture (Galileo and Einstein)   Link to lecture (Physics 152)