Alien Earths – Using Conic Sections to Explore the Solar System

Teachers: Amena Mitha and Andy Eschbacher

Target Audience: 11th Grade Algebra II and Physics students

Project Description:

The Alien Earths project is a multi-disciplinary project-based lesson series that encompasses concepts in Algebra II about conic sections, the dynamics of gravity in the solar system, and a search for potential life to motivate a space mission. The lessons will take place within a 6-week period that culminates with a final project that pulls in the previous lessons in an interactive and technologically motivated way. The lesson sequence is as follows:

  1. Alien Earths – Search for life – an inquiry-driven lesson that lets students explore the solar system’s planets, moons, and other objects for properties that may support life;
  2. Introduction to Gravity – an interactive and generative lesson that allows students to use computer simulations as a playground to explore the shapes and dynamical quantities involved in gravity in our solar system;
  3. Conic Sections – another interactive simulation that allows students to discover different types of conic sections;
  4. Centripetal Force Lab – a lab exploration of central forces that uses everyday objects to explore some of the dynamics of gravity;
  5. Gravity Analysis – a more in-depth lesson on gravity that draws from the experiences of the two previous lessons to allow students to explore and find empirical data for the equations that govern gravity;
  6. Orbit of Satellites – a lesson based on a NASA applet (via the internet) that uses real-time data of satellite positions and trajectories that allows students to explore the shape of satellite orbits in tangible and relevant situation;
  7. Stop-Motion Animation -- An animation that represents the culmination of the work of the whole project

Driving Question:

The driving question for the Alien Earths project is, “How can we use conic equations to plan a mission to explore a planet or moon to find life?” Within this driving question there are three lesson objectives, with the question stressing that all the objectives are intimately intertwined. The first, on the use of conic sections, is a part of mathematics education that is rarely discussed in detail and is usually motivated from a decontextualized point of view. Next, the part of the question that deals with gravity is implicit in ‘plan a mission to explore a planet or moon’. Gravitation of varying acceleration is a difficult subject to ‘explore’ in a classroom setting since humans experience the same acceleration for most heights. We do not have practical experience experimenting with gravity as one would have, say, experience experimenting with merry-go-rounds or friction. Therefore, developing an intuition from the ground up is a major objective that the project aims to address.

Overall Goals:

The goals of the project include: gaining intuition about planetary and satellite orbits, understanding the similarities and differences between various conic sections and their applications to science, and, finally, illustrating the mechanisms of gravity and how it affects objects in space.

Project Objectives:

The Alien Earths Project consists of seven integrated lessons, each with unique student objectives. In this project, students will be able to:

Alien Earths

  1. Identify characteristics for environments that are suitable for life.
  2. Conduct research using the internet to identify planets and/or moons that are potential candidates for further exploration.

Introduction to Gravity

  1. Describe the orbits of planetary bodies
  2. Identify the two independent variables (mass and distance) that determine the magnitude of a gravitational force
  3. Develop an intuition about celestial dynamics

Conic Sections

  1. Relate representations of quadratic functions, such as algebraic, tabular, graphical and verbal descriptions.
  2. Identify the similarities and differences between different types of conic sections.
  3. Recognize various planetary orbits and relate them to conic sections

Centripetal Force

  1. Compare measurements to theoretical results
  2. Relate centripetal forces from tension to centripetal forces due to gravity

Gravity Analysis

  1. Compare measurements to theoretical results
  2. Relate centripetal forces from tension to centripetal forces due to gravity

Orbit of Satellites

  1. Calculate forces from Newton’s law of gravitation
  2. Infer functional forms through data interpretation
  3. Identify forces and velocity on a free-body diagram of a planet in a circular orbit

Stop-Motion Animation

  1. Reinterpret dynamics into a frame-by-frame animation
  2. Construct conic section equations that intersect at specific locations

Rationale:

This project will connect math and physics concepts to real-world applications. This is important because often times the connections between high school subjects are too tenuous. Further, the subject of space exploration is very rich and leaves a wide range of people with a sense of wonder. With all the space missions to Mars, Jupiter, and Saturn as well as their moons, it is easy to keep the project relevant and include recently discovered phenomena. The project contains a wide variety of activities that will appeal to all learning styles, which is something we believe will maximize participation, learning, and creativity.

Background:

Major concepts addressed in the lessons (see separate sheet Concept Map as well):

1.  Forces

  1. Gravitational forces – as a cause of planetary orbits;
  2. Newton’s second law – a law that relates force, mass, and acceleration;
  3. Centripetal forces – ‘center seeking’ forces responsible for allowing one to stay in an orbit or to negotiate a tight turn in a car.

2.  Conic Sections

  1. Ellipse – shapes that describe the shape of stable planetary orbits for the case when the mass of the orbiting body is much smaller than the mass of the gravitating center;
  2. Hyperbola – important shapes for describing trajectories that escape planetary orbits;
  3. Parabola – the limiting case between ellipses and hyperbolae. The form y = ax2+bx+c is important in mathematics (e.g., finding the roots of equations) and science (e.g., near-earth dynamics).

3.  Data interpretation – using data to infer the properties of an ellipse

4.  Viability of life-sustaining worlds – biological investigations into requirements of life (water, nutrients, energy source, etc.)

Standards:

Biology:

11. Science Concepts. The student knows that organism maintain homeostasis. The student is expected to:

b. Investigate and identify how organisms, including humans, respond to external stimuli;

12. Science Concepts. The student knows that interdependence and interactions occur within an ecosystem. The student is expected to:

b. Interpret interactions among organism exhibiting predation, parasitism, commensalism, and mutualism;

d. Identify and illustrate that long-term survival of species is dependent on a resource base that may be limited;

Physics:

1.  Scientific processes. The student, for at least 40% of instructional time, conducts field and laboratory investigations using safe, environmentally appropriate, and ethical practices. The student is expected to:

a.  Analyze examples of uniform and accelerated motion including linear, projectile, and circular;

b.  Demonstrate the effects of forces on the motion of objects.

4.  Science Concepts. The student knows the laws governing motion. The student is expected to:

b.  Analyze examples of uniform and accelerated motion including linear, projectile, and circular;

c.  Demonstrate the effects of forces on the motion of objects;

6.  Science Concepts. The student knows forces in nature. The student is expected to:

a.  Identify the influence of mass and distance on gravitational forces;

Algebra II:

5. Algebra and geometry. The student knows the relationship between the geometric and algebraic descriptions of conic sections. The student is expected to:

a. Sketch graphs of conic sections to relate simple parameter changes in the equation to corresponding changes in the graph;

b. Identify symmetries from graphs of conic sections;

c. Identify the conic section from a given equation;

7. Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student is expected to:

a. Use characteristics of the quadratic parent function to sketch the related graphs and connect betweenthey=ax2+bx+cand they=a(x-h)2+ksymbolic representations of quadraticfunctions;

b. Use the parent function to investigate, describe, and predict the effects of changes ina,h, andkon the graphs ofy=a(x-h)2+kform of a function in applied and purely mathematical situations;

Final Product:

The final product will be as follows: Students are to create a stop-motion animation that takes a spacecraft from an earth orbit of their choosing (elliptical or circular) to a hyperbolic orbit that intersects the planet and finally maintains a stable orbit. Students are expected to respect distance and time scales as well as relevant motion of the planets/moons as the spacecraft is on its mission. For instance, if a spacecraft is to travel from earth to mars, mars is not at the future point at which the spacecraft will intersect it (mars), so mars must move into to the position in a relevant amount of time.

This project encompasses the students’ knowledge of conic sections as descriptions of planetary orbits, reveals an alternate interpretations of ‘velocity’ through a transformation to a discrete time, and how one can interconnect different conic sections to form a rough trajectory of what a spacecraft would follow going from the earth to another planet or moon.