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Kepler's First Law

Kepler's First Law . The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. . The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). . Kepler's Second Law:.

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Kepler's First Law

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  1. Kepler's First Law The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse).

  2. Kepler's Second Law: II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun.

  3. Kepler's Third Law: III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes: In this equation P represents the period of revolution for a planet and R represents the length of its semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semimajor axes for the two planets are assumed to be in the same distance units.

  4. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.

  5. What Really Happened with the Apple? The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. Thus, by Newton's 2nd Law there must be a force that acts on the apple to cause this acceleration. Let's call this force "gravity",

  6. Sir Isaac's Most Excellent Idea Now came Newton's truly brilliant insight: if the force of gravity reaches to the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon!

  7. If we increase the muzzle velocity of an imaginary cannon, the projectile will travel further and further before returning to earth. Newton reasoned that if the cannon projected the cannon ball with exactly the right velocity, the projectile would travel completely around the Earth, always falling in the gravitational field but never reaching the Earth, which is curving away at the same rate that the projectile falls. That is, the cannon ball would have been put into orbit around the Earth. Newton concluded that the orbit of the Moon was of exactly the same nature

  8. the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit.

  9. By such reasoning, Newton came to the conclusion that any two objects in the Universe exert gravitational attraction on each other, with the force having a universal form:

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