“In it [differential calculus], mathematics becomes a dynamic mode of thought, and that is a major step in the ascent of man,” wrote Dr. Jacob Bronowski in The Ascent of Man. In a previous post, we saw how differential calculus gives us a dynamic mode of thought. Because static graphs in a textbook fail to capture a feel for the dynamics of a model, in this post, we’ll discover how computer graphing technology can be used to create animations that demonstrate the underlying dynamics in a mathematical model of a physical system.
For the first demonstration, a polar equation of an ellipse is used to model Kepler’s first and second laws of planetary motion. The polar equation r = 0.5*20 / (1 + 0.5Cos(θ)) has eccentricity = 0.5, and foci at (0, 0) and (13.333, 1800). Kepler’s first and second laws of planetary motion are given below. The Earth’s elliptical orbit has eccentricity = 0.0167 which results in an almost circular orbit. This is probably why Copernicus thought that the planets traveled at a constant speed in circular orbits around the Sun. Saturn’s elliptical orbit has eccentricity = 0.0556 which results in the familiar oval shape of an ellipse.
First Law: All planets in the solar system orbit the Sun on an elliptical curve where the Sun is located at one of the focus points of the ellipse.
Second Law: The speed of a planet increases as the planet moves closer to the Sun, and decreases as the planet moves farther from the Sun. A line segment joining a planet and the Sun sweeps out equal areas during equal time intervals.
In the animation below, the Sun is located at polar point (13.333, 1800), and the moving trace mark represents a planet orbiting the Sun. The two sectors, marked with red segments, represent equal area sectors that were swept out in equal time intervals. Because the software uses the origin and the polar trace mark point on the curve to draw the radius of a polar trace mark, it may appear that a planet is primarily orbiting about the second focus point instead of the Sun.
The second demonstration uses the polar equation r = 20 to model a terrifying gut wrenching ride on a Ferris wheel that has a 40-foot diameter, and turns counterclockwise one revolution every 12 seconds. The moving trace mark represents a rider’s position at time t in seconds, and t = 0 seconds when the angular position of the rider = 0 degrees. Using differential calculus, the rider’s horizontal velocity and vertical velocity at time t can be deduced. Refer to the table below. Anyone who has ridden on a Ferris wheel remembers the forces acting on his/her body as the result of his/her changing horizontal and vertical velocity as the wheel turns.
Teaching Points: (Of course, what is taught depends on the mathematical level of the student.)
- We live in a dynamic, changing world. Students should be exposed to the concept of variable rate of change as early as possible. Even though younger students can’t do differential calculus, a teacher-directed animation of the rate concept will help students better understand how mathematics can describe some of nature’s laws.
- Students should be taught that linear relationships are characterized by a constant rate of change. The dependent variable changes at a constant rate with respect to the independent variable. Students can see the constant rate of change of the vertical velocity of the trace mark as the trace mark advances left to right on the graph of a line.
- Show students the movement of a trace mark on the curve y = 8Sin(x). Because the x-variable is changing at a constant rate, the horizontal velocity of the trace mark is constant. All students can see that the vertical velocity of the trace mark changes as the trace mark advances left to right. If they imagine that they are on a roller coaster, they can feel the variable forces acting on their body as the trace mark advances left to right.
- Show students the movement of a trace mark on the curve y = 3x^(1/3). Where the curve is somewhat linear, the vertical velocity of the trace mark is almost constant. As the trace mark approaches the origin, the vertical velocity of the trace mark increases. It can be explained to a calculus student why the vertical velocity of the trace mark is infinite for a fleeting instant of time x = t = 0. Because the x-variable is changing at a constant rate, the x-variable can be treated as time variable t.
- The Johannes Kepler (1571-1630) and Tycho Brahe (1546-1601) relationship is an interesting story. Day and night for many years, in his observatory on the island of Hven, near Copenhagen, Tycho Brahe carefully recorded the positions of the planets and stars. In 1600 Kepler met Tycho Brahe, and gained access to Brahe’s data. In the nine-year period after Tycho Brahe’s death, Kepler used the observational data to deduce his first and second laws of planetary motion. Kepler discovered his third law of planetary motion much later. It is difficult to understand, imagine, or appreciate how Kepler was able to use inductive reasoning to discover the patterns of planetary motion. Students should be told this story because it demonstrates the monumental gift of human intelligence, and the struggle that is required to advance knowledge.
The graphics in this post were created with the program, Basic Trig Functions, which is offered by Math Teacher’s Resource. In addition to graphing x-y variable relations and polar functions, users can graph the powers or roots of a complex number, and view a list of the powers or roots, which appears to the right of the graphic output. Segments and vectors can be drawn by left-clicking and dragging the mouse. The Edit/ Edit Graphics menu provides options for setting segment color, pen width, and Head/Tail parameters.
The user interface for all program modules is simple and intuitive. When graphing equations, users can select a sample equation, which is automatically pasted into the active equation edit box. When appropriate, the program provides comments and suggestions for setting screen parameters to achieve best results. After an equation is graphed, you can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. With simple mouse control clicks, you can find relative minimum points, relative maximum points, x-intercepts, and intersection points. A Help menu provides a quick summary of all magical mouse control clicks. Of course, all graphs can be copied to the clipboard and pasted into another document. To view multiple screen shots of the program’s modules, go to www.mathteachersresource.com. Click the “learn more” button in the TRIGONOMETRIC FUNCTIONS section. Teachers will find useful comments at the bottom of each screen shot.