5 Minute Lesson: Kepler’s 2nd Law

Instructor: Mr. Nalin Ratnayake
Subject: Physics (algebra-based)
Target Grade Level: 11/12
Lesson Objective: Understand the major implication of Kepler’s 2nd Law.

Good morning class. Have you ever wondered about the motion of the planets? My name is Mr. Ratnayake (Mr. Rat will do). Today we will discuss Kepler’s 2nd Law of Planetary Motion, building on your previous knowledge of basic mechanics, algebra, and geometry.

In the early 1600’s, most astronomers believed that if the planets orbited around the sun, they must do so in circles. However, astronomical observations did not agree; planets seemed to move randomly in the sky. It was a mystery. A German mathematician named Johannes Kepler forever changed astronomy by demonstrating in his *first* law of planetary motion that by simply treating an orbit as an ellipse, instead of the previously-assumed circular shape, the simplicity and harmony of planetary motion became clear. He then went on to explain the consequences of elliptical motion in his *second* law, which states: A line connecting a planet to the sun sweeps out equal areas in equal times.

Refer to the diagram on your handout, or follow me on the board. Suppose I have here my orbital ellipse, and I consider the area swept out by the radius r in some interval of time. The planet has moved on an arc, by an angle θ as measured from the sun. Now we have set up our problem and will commence, like good scientists, to ask questions.

What basic geometric shape does this look like? (Triangle). Let me draw this triangle. Who knows how to find the area of a triangle? (one-half the base times the height). Ok. Do we have a variable on our diagram that looks like it would be the height of the triangle?  (the radius r).  And this arc forms the base. What is the length of an arc (You remember from geometry of course, it is the radius of the arc times the subtended angle.)  So the area (A) of this pie wedge is…. ½ (r θ) r , or ½ r2θ.

Let’s assume, as Kepler did, that this area must remain constant for the same time interval in an orbit. Consider our diagram. If our planet moved closer to the sun, say here, then the radius, our distance to the sun, is much smaller. To maintain the same area of triangle, what must happen to its base? (must get larger). Remember, this is the arc length we traveled in our orbit for a set time interval. We traveled farther in the same amount of time than we did out here! What can we deduce about our speed? (we went faster!). We move faster when we are near the sun on our orbit and slower when we are far from the sun on our orbit.

Was Kepler right? It turns out that his theory exactly matched observational data from astronomer Tycho Brahe; this explained previously erratic motions of the heavenly bodies with a simple concept. Today, Kepler’s Laws enable us to predict the motion of the planets, asteroids, comets, and satellites in space. NASA’s Mars Curiosity probe just launched this week. It will take 8 months to reach Mars. Thanks to Kepler’s Laws we know exactly where Mars will be and how fast it will be going in 8 months; and Curiosity will be in the right place at the right time.

Check for understanding, all together now, and I’m looking for every one of you to answer. As a planet gets closer to the sun, does it speed up or slow down? (Speeds up!) As a planet gets further away on its orbit, does it speed up or slow down? (Slows down!) Good. Any questions?

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