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ELLIPSES

ELLIPSES. Practical Aplications. Ellipses are used in studying natural phenomena in several areas. It was Johannes Kepler (1571 – 1630) who first discovered that the planets didn’t orbit the sun in true circular orbits but rather in elliptical orbits.

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ELLIPSES

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  1. ELLIPSES Practical Aplications

  2. Ellipses are used in studying natural phenomena in several areas. It was Johannes Kepler (1571 – 1630) who first discovered that the planets didn’t orbit the sun in true circular orbits but rather in elliptical orbits. Johannes Kepler was a scientist who studied under another scientist named Tycho Brahe. Brahe was renowned for his highly precise charting of the stars and planets using rudimentary instruments. Kepler used Brahe’s data to describe the motion of the planets. Kepler concluded that if Brahe’s data was precise, Mars’ orbit could not be a circle because it was off by 8 arcminutes.

  3. Kepler was forced to abandon uniform circular motion and concluded Mars' orbit was not a circle but an ellipse. He published the results of his work in 1609. He devised three laws of planetary motion, the first of which states that “The orbits of planets are ellipses with the sun at one focus.” Ellipses are characterized by two numbers: Semimajor Axis: (a) size of the longest axis Example: The Orbit of Mars: a = 1.5237 AU; e = 0.0934 Eccentricity: (e = c/a) - shape of the ellipse.

  4. The point on the orbit where the planet is nearest the Sun is called the perihelion. Aphelion is where the planet is farthest from the Sun. For our planets, the foci are very near the center and one focus point is the Sun. Therefore, the ‘c’ distances are very small. If ‘c’ were 0, the ellipse would actually be a circle. c = (152 500 000 – 147 500 000)/2 = 2 500 000 km a = (152 500 000 + 147 500 000)/2 = 150 000 000 km

  5. Kepler’s 2nd Law of Planetary Motion The line joining the Sun and the planet sweeps out equal areas in equal times. • Planets move fastest at Perihelion (closest to the Sun) • Planets move slowest at Aphelion (farthest from the Sun) • The Second Law provides a geometric description of the change in speed that completely eliminates ideas used by older theories. It was previously believed that the speed of orbiting planets remained constant.

  6. Whereas a planet’s orbit is very nearly a circle, a comet’s orbit is a highly eccentric ellipse.

  7. When the exact landing location is uncertain the probable landing area is shaped like an ellipse. The Beagle 2 Landing Site: Viking Orbiter image mosaics of the landing site region. The colored ellipses present the areas in which Beagle 2 may land.

  8. There are some large halls that are designed so that the ceiling has an ellipsoid (three-dimensional ellipse) shape. This means that two people, each standing at a focus and separated by a large distance can have a private conversation with each other without those people standing between them hearing.

  9. V3 (0,13) F1 (0,5) V2 (-12,0) V1 (12,0) F2 (0,-5) V4 (0,-13) a2 = 144 a = 12 b2 = 169 b = 13 V1 (12,0) V2 (-12,0) V3 (0,13) V4 (0,-13) c2 = b2 - a2 c2 = 169 – 144 c2 = 25 c = 5 F1 (0,5) F2 (0,-5)

  10. V3 (0,6) V2 (-10,0) F1 (8,0) V1 (10,0) F2 (-8,0) V4 (0,-6) a2 = 100 a = 10 b2 = 36 b = 6 V1 (10,0) V2 (-10,0) V3 (0,6) V4 (0,-6) c2 = a2 - b2 c2 = 100 – 36 c2 = 64 c = 8 F1 (8,0) F2 (-8,0)

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