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Understanding Newton's Mountain

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

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)