GOD & MATH: Thinking Christianly About Math Education



Kepler’s Laws of Planetary MotionJohannes Kepler (1571 – 1630)Born in Weil der Stadt, Württemberg, Holy Roman Empire (now Germany)Johannes Kepler was a German mathematician and astronomer who discovered that the Earth and planets travel about the sun in elliptical orbits. He gave three fundamental laws of planetary motion. He also did important work in optics and geometry. Kepler’s main written work is called Harmonices mundi libri, or The Harmony of the World (1619)Kepler’s Three Laws of Planetary MotionLaw I (the Ellipse Law) - the curve or path of a planet is an ___________________ with the Sun fixed at one _______________. Law II (the Area Law) - the ______________________ taken by a planet to reach a particular position is represented by the ___________________ swept out by the radius vector drawn from the fixed Sun.Law III (the Time-Radius Law) - for any two planets the ratio of the squares of their _________________ will be the same as the ratio of the cubes of the mean ________________ of their orbits. Investigate Kepler’s Laws by working through the following questions, based on this website: First Law:1. Check all the boxes under “Kepler’s first law.” Set parameters for Mercury, and hit “Start animation.” What fraction of the semi-major axis is the focal radius (just estimate by looking)? What is the eccentricity of Mercury?2. Repeat question 1 for Mars, Uranus, and Pluto (you may need to adjust the animation rate for the outer planets).Mars:Uranus:Pluto: 3. Manually change the eccentricity of the orbit to 0.5. What fraction of the semi-major axis is the focal radius?4. Manually change the eccentricity of the orbit to 0.7. What fraction of the semi-major axis is the focal radius?5. Suggest a formula for eccentricity, and describe (in a complete sentence) how the eccentricity of an ellipse affects its shape. 6. What is the earth’s eccentricity, and what does this tell us about the shape of the earth’s orbit?Kepler?s Second Law:1. Un-check all the boxes under “Kepler’s first law.” Select the “2nd law” tab, and leave your eccentricity at 0.7 (a fictional planet). Adjust the “sweep size” to 1/20 (which means 20 sweeps would make one full revolution for this planet), and start sweeping. Perform at least 4 non-overlapping sweeps.a. Describe the “shape” of the sweep as it relates to the distance of the planet from the sun.b. Complete the statement: When a planet is far from the sun, it travels (faster/slower) __________________________ than when it is close to the sun. 2. Set parameters for Mercury, being sure that the “sweep size” is still 1/20. Sweep several times. Record the sweep duration in earth years, and the sweep area in square astronomical units (one AU is equal to the average distance from the earth to the sun)sweep duration:sweep area:3. Repeat question 2 for Earth, Jupiter, and Neptune:EarthJupiterNeptunesweep duration:sweep area: 4. Explain the differences in sweep duration and sweep area in relation to a planet’s distance from the sun.Kepler?s Third Law:If T is the period (time) for one complete orbit of a planet, and R is the average distance (radius) of the planet from the sun, then the following equation is another form of this law: 1. Solve this equation for the period.2. This means that “the period of a planet varies ______________________________ with the __________________ power of the average distance from the sun.”3. Select the “3rd law” tab, and “linear plot.” Under “visualization options,” select “show solar system orbits.” The graph of T vs. A is a (curve / line) ________________________.4. Set parameters for earth. What are the coordinates for earth? ____________________ Explain (in a complete sentence) why this makes sense. 5. Now select “logarithmic plot” and describe the appearance of the plot.6. Starting with the equation , take the common log of both sides, and simplify completely, using the laws of logarithms (remember: “k” is a constant). How does this equation connect to your answer to #5?Your turn: Suppose the following parametric equations represented the path of an actual planet, where one unit represents 1 AU (93 million miles, 149.6 million km): T0xyDistancefrom sun1. Complete the (x, y) rows of the table above (use a calculator and round to 3 decimal places), and plot each of the corresponding points on the coordinate axis system below. 2971800869952. Convert the parametric equations to rectangular (x, y) form using the Pythagorean Trigonometric identity.Use this form to show that the sun (one of the foci) is at (4, 0).3. On the graph above, draw the line segments connecting each (x, y) location to the sun. Shade three distinct “sweeps.”4. Complete the “distance” row in the table above, indicating the distance of the planet (in millions of kilometers) from the sun at each of the given times. Show your work for T = only, and then let your calculator (List editor, or “2nd-Enter”) help you repeat the process efficiently. 5. Use the distance data to find a reasonable approximation to the average distance of this planet from the sun (am I being mean here?). 6. What is the period for this planet? (Hint: )7. Using your answers to #s 5 and 6, find the constant of variation from Kepler’s third law. (please note: this equation does not reflect an actual planet!) Kepler Quotes: “…I give thanks to Thee, O Lord Creator, Who hast delighted me with Thy makings and in the works of Thy hands have I exulted…I have made manifest the glory of Thy works, as much of its infinity as the narrows of my intellect could apprehend.” Question: What can you learn about Kepler’s beliefs based on the above quote? ”I feel carried away and possessed by an unutterable rapture over the divine spectacle of the heavenly harmony.” Question: Based on what you know of Kepler’s laws, why would he use the words “heavenly harmony”? “I consider it a right, yes a duty, to search in cautious manner for the numbers, sizes and weights, the norms for everything He has created. For He himself has let man take part in the knowledge of these things and thus not in a small measure has set up His image in man...”Restate this quote in your own words:What is the connection between studying mathematics and the image of God in man?Kepler’s goal as a scientist/mathematician was “…to make “as many discoveries as possible for the glorification of the name of God and sing unanimous praise and glory to the All-Wise Creator.”Do you think a scientist/mathematician in the 21st century can have the same goal? Explain your thinking. ................
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