This lesson challenges students to apply their knowledge of object motion by animating sequences of hand-rendered pictures that model a set of physical conditions. The challenges include animating the orbital motion of planets and satellites, the effects of gravity on a falling body, and motions of objects in inertial (moving) frames of reference. The lesson was created by a high school physics teacher to help learners build quantitative reasoning skills in preparation for understanding kinematics.

Editor's Note:Modeling is a powerful way for students to relate the math formula to the physical process under study. This lesson allows learners to develop hand-crafted "flipbook" models of motion before they advance to computer modeling. In each challenge, data is provided so the animations can be computationally accurate.

All positions of objects and the directions of forces and motions must be described in an arbitrarily chosen reference frame and arbitrarily chosen units of size. In order to share information with other people, these choices must also be shared. (6-8)

Crosscutting Concepts (K-12)

Systems and System Models (K-12)

Models can be used to represent systems and their interactions. (6-8)

Models (e.g., physical, mathematical, computer models) can be used to simulate systems and interactions—including energy, matter, and information flows—within and between systems at different scales. (9-12)

NGSS Science and Engineering Practices (K-12)

Developing and Using Models (K-12)

Modeling in 6–8 builds on K–5 and progresses to developing, using and revising models to describe, test, and predict more abstract phenomena and design systems. (6-8)

Develop a model to predict and/or describe phenomena. (6-8)

Modeling in 9–12 builds on K–8 and progresses to using, synthesizing, and developing models to predict and show relationships among variables between systems and their components in the natural and designed worlds. (9-12)

Develop and use a model based on evidence to illustrate the relationships between systems or between components of a system. (9-12)

AAAS Benchmark Alignments (2008 Version)

2. The Nature of Mathematics

2A. Patterns and Relationships

9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

2C. Mathematical Inquiry

9-12: 2C/H2. Much of the work of mathematicians involves a modeling cycle, consisting of three steps: (1) using abstractions to represent things or ideas, (2) manipulating the abstractions according to some logical rules, and (3) checking how well the results match the original things or ideas. The actual thinking need not follow this order.

4. The Physical Setting

4F. Motion

6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.

6-8: 4F/M3b. If a force acts towards a single center, the object's path may curve into an orbit around the center.

9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.

9-12: 4F/H2. All motion is relative to whatever frame of reference is chosen, for there is no motionless frame from which to judge all motion.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

4G. Forces of Nature

6-8: 4G/M1. Every object exerts gravitational force on every other object. The force depends on how much mass the objects have and on how far apart they are. The force is hard to detect unless at least one of the objects has a lot of mass.

6-8: 4G/M2. The sun's gravitational pull holds the earth and other planets in their orbits, just as the planets' gravitational pull keeps their moons in orbit around them.

11. Common Themes

11B. Models

9-12: 11B/H1a. A mathematical model uses rules and relationships to describe and predict objects and events in the real world.

9-12: 11B/H5. The behavior of a physical model cannot ever be expected to represent the full-scale phenomenon with complete accuracy, not even in the limited set of characteristics being studied. The inappropriateness of a model may be related to differences between the model and what is being modeled.

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

MP.4 Model with mathematics.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^{?}

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

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The Animating Motion activity provides an engaging project that coordinates with topics in the One-Dimensional Motion chapter of The Physics Classroom Tutorial.