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Orbits and conic sections

What are conic sections?

conic section

All orbits turn out to be parts of a conic section!

Depending on how you launch a satellite into space, your spaceship can have an orbit with any of these shapes.

Circular orbits

Elliptical orbits

Parabolic orbit

Hyperbolic orbit

What is a hyperbola? What is an asymptote?

http://www.mathsisfun.com/geometry/hyperbola.html

Satellite Motion Chap 14 Hewitt

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Hyperbola’s velocity at infinity.

http://hopsblog-hop.blogspot.com/2013/03/what-heck-is-vinf.html

math

Math lines We Know Drama Asymptotes

 

Learning Standards

2016 Massachusetts Science and Technology/Engineering Curriculum Framework

8.MS-ESS1-2. Explain the role of gravity in ocean tides, the orbital motions of planets, their moons, and asteroids in the solar system.

HS-ESS1-4. Use Kepler’s laws to predict the motion of orbiting objects in the solar system.
Describe how orbits may change due to the gravitational effects from, or collisions
with, other objects in the solar system.

HS-PS2-4. Use mathematical representations of Newton’s law of gravitation and Coulomb’s law to both qualitatively and quantitatively describe and pre

HS-PS2-10(MA). Use free-body force diagrams, algebraic expressions, and Newton’s laws of motion to predict changes to velocity and acceleration for an object moving in one
dimension in various situations.

Massachusetts Curriculum Framework for Mathematics
Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a conic section.
MA.3.a. (+) Use equations and graphs of conic sections to model real-world problems.
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

A Framework for K-12 Science Education: Practices, Crosscutting Concepts, and Core Ideas (2012)

PS2.B: TYPES OF INTERACTIONS

Gravitational, electric, and magnetic forces between a pair of objects do not require that they be in contact. These forces are explained by force fields that contain energy and can transfer energy through space. These fields can be mapped by their effect on a test object (mass, charge, or magnet, respectively). Objects with mass are sources of gravitational fields and are affected by the gravitational fields of all other objects with mass. Gravitational forces are always attractive. For two human-scale objects, these forces are too small to observe without sensitive instrumentation. Gravitational interactions are non-negligible, however, when very massive objects are involved. Thus the gravitational force due to Earth, acting on an object near Earth’s surface, pulls that object toward the planet’s center. Newton’s law of universal gravitation provides the mathematical model to describe and predict the effects of gravitational forces between distant objects. These long-range gravitational interactions govern the evolution and maintenance of large-scale structures in the universe (e.g., the solar system, galaxies) and the patterns of motion within them… Newton’s law of universal gravitation and Coulomb’s law provide the mathematical models to describe and predict the effects of gravitational and electrostatic forces between distant objects.

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