Pre-CalcFreebersyser

Block 3

Conic Get Any Better than 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.

Why an Ellipse?

Equation Major Axis Vertices Co-Vertices=1 Horizontal (± , 0) (± , 0)

=1 Vertical (0, ± ) (0, ± ) Foci equation:= -

Equation

Ellipse- set of all points P such that sum of distances between P and two fixed points (foci) is constantFoci- two fixed points in an ellipseVertices- points at which line through foci intersects ellipseMajor axis- line segment that joins verticesCenter- midpoint of major axisCo-vertices- points of intersection of ellipse and line perpendicular to major axis at centerMinor axis- line segment that joins co-vertices

Vocabulary

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 1609Triangulating the positions of the planets then using the ellipse equation to figure out that the circular model for planetary orbit was inaccurateEach planet orbits around two fixed points of gravitation (foci) which causes their rotation to be elliptical

Planetary Ellipse

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 empty2. 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 pull3. 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

Kepler’s Three Laws of Planetary Motion

The equation for the ellipse allowed Kepler to come up with his planetary theoryHad 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

Coming Full “Ellipse”

Bike chain moves around two fixed points to allow motionPulleys also use the idea of an ellipse to allow the forward motion of heavy loadsUse of ellipses in physics to figure out forward motionThe “skip it” toy that attaches to one ankle while you skip over it with the other leg in an elliptical motion

Other Ellipses in the Real World

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

Works Cited