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Conic Get A ny B etter than an Ellipse?

Conic Get A ny B etter than an Ellipse?. Pre- Calc Freebersyser Block 3. Why an Ellipse?. I chose the ellipse as my conic section to study because we briefly learned about it in physics when we studied planetary orbits, but I wanted to learn more in depth about them. Equation.

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Conic Get A ny B etter than an Ellipse?

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  1. Conic Get Any Better than an Ellipse? Pre-Calc Freebersyser Block 3

  2. Why an Ellipse? • I chose the ellipse as my conic section to study because we briefly learned about it in physics when we studied planetary orbits, but I wanted to learn more in depth about them.

  3. Equation Equation Major Axis Vertices Co-Vertices =1 Horizontal (±, 0) (±, 0) =1 Vertical (0, ±) (0, ±) Foci equation: = -

  4. Vocabulary • Ellipse- set of all points P such that sum of distances between P and two fixed points (foci) is constant • Foci- two fixed points in an ellipse • Vertices- points at which line through foci intersects ellipse • Major axis- line segment that joins vertices • Center- midpoint of major axis • Co-vertices- points of intersection of ellipse and line perpendicular to major axis at center • Minor axis- line segment that joins co-vertices

  5. Planetary Ellipse • Kepler first discovered that the orbit of Mars could not be a circle but a type of oval rotation, thus coming up with the theory of planetary motion, in 1609 • Triangulating the positions of the planets then using the ellipse equation to figure out that the circular model for planetary orbit was inaccurate • Each planet orbits around two fixed points of gravitation (foci) which causes their rotation to be elliptical

  6. Kepler’s Three Laws of Planetary Motion • 1. Planets move in ellipses with the Sun at one focus • "all planets move in elliptical orbits with the Sun at one focus and the other focus empty“ • The same concept is used with satellites, the earth as one foci and the other is empty • 2. The radius vector describes equal areas in equal times • "the line joining the planet to the Sun sweeps over equal areas in equal time intervals“ • When the satellite comes closer to the center of the earth its speed increases because of earth’s gravitational pull • 3. Squares of periodic times are to each other as cubes of the mean distances • "For any planet, the square of its period of revolution is directly proportional to the cube of its mean distance from the Sun“ • Explains that the farther away a satellite is from the earth the longer the satellite takes to complete an orbit

  7. Coming Full “Ellipse” • The equation for the ellipse allowed Kepler to come up with his planetary theory • Had Kepler not used that equation and point triangulation, then today we would still think that the planets have circular orbits, and that the Earth is the center of our solar system instead of the Sun

  8. Other Ellipses in the Real World • Bike chain moves around two fixed points to allow motion • Pulleys also use the idea of an ellipse to allow the forward motion of heavy loads • Use of ellipses in physics to figure out forward motion • The “skip it” toy that attaches to one ankle while you skip over it with the other leg in an elliptical motion

  9. Works Cited • http://www.thisiscarpentry.com/wp-content/uploads/2012/03/Ellipse-Diagram_2.jpg • http://www.k12.hi.us/~mathappl/MAch3Curves.html • http://www2.norwalk-city.k12.oh.us/wordpress/halgebra20708/files/2008/05/planet_solarsystem.gif • http://www.keplersdiscovery.com/Elipse.html • http://www.deeringmath.com/Conics/images/conics.png • http://space.about.com/od/astronomybasics/tp/Keplers_Three_Laws.htm • http://hipcycle.com/media/wysiwyg/bike_chain.gif • http://visual.merriam-webster.com/images/science/physics-mechanics/double-pulley-system.jpg • http://patentimages.storage.googleapis.com/EP0384953A1/imgf0001.png • http://upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Angular_Parameters_of_Elliptical_Orbit.png/200px-Angular_Parameters_of_Elliptical_Orbit.png • http://www.ancient-world-mysteries.com/images/Image10a.gif • http://www.keplersdiscovery.com/Images/Keplers.Ch24%20Diagram.jpg • http://www.mathwarehouse.com/ellipse/images/translations/translation_0_2.gif

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