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Cannon on a Mountain

Here we're animating a diagram Isaac Newton drew himself, and published in his book A Treatise of the System of the World, page 5.

He placed a huge cannon atop an imaginary mountain, and therefore, of course, far above the atmosphere, so no worries about air resistance.

For low speeds, the cannonball doesn't get far relative to the Earth's size, so the downward gravity gives the usual parabolic path (to an excellent approximation). At higher speeds, the cannonball goes far enough that the direction of gravity—always toward the center of the Earth—changes noticeably during flight, and therefore so does the shape of the path!

At a certain speed, the cannonball goes all the way round. Check that, then explore what happens at even greater speeds.

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