Download Conic Get Any Better 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
Equation
𝑥2
𝑎2
𝑥2
𝑏2
Major Axis
Vertices
Co-Vertices
+
𝑦2
=1
2
𝑏
Horizontal
(± 𝑎, 0)
(± 𝑏, 0)
+
𝑦2
=1
𝑎2
Vertical
(0, ± 𝑎)
(0, ± 𝑏)
Foci equation:
𝑐 2 = 𝑎 2 -𝑏 2
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
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
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
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
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
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.norwalkcity.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