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31 October 2014

Are Kepler's Laws part of AP Physics 1? No and Yes.

Debby Heyes, who attended my open lab this summer,has a quick AP physics 1 question:

Are Kepler's Laws included in the course?  

Fast answer: no.  A search in the curriculum guide for "Kepler" gives no results.

Deeper answer: Yes AND No.  Kepler's laws by name are not part of the curriculum, but some of the behavior of planetary orbits described by Kepler's laws is part of AP Physics 1.

The "equal areas / equal times" law can be stated as a consequence of angular momentum conservation.  An orbiting planet experiences no torque relative to the central star (because the gravitational force always points back to the center of rotation, meaning the distance term in "torque = force * distance" is zero).  Therefore, the planet's angular momentum about the central star is conserved.  Treating the planet as a point object, its angular momentum is given by mvr, where r is the distance to the central star.  When r goes down -- i.e. when the planet is closer to the sun -- v goes up, meaning the planet moves faster in its orbit.  That's essentially Kepler's law.

The "period proportional to the 3/2 power of the radius" law is merely a consequence of Newton's second law and circular motion, at least if we consider circular orbits only (which we emphatically do in AP Physics 1).  Set the gravitataional force equal to ma, where the acceleration in circular motion is v2/r.  Then the speed of an object in circular motion is the circumference divided by the orbital period.  Solving for period gives the Kepler's law relationship -- and we should be able to do that and understand it in AP Physics 1.

The law that says "all orbits are ellipses with the sun at one focus" is not in any way on the AP Physics 1 exam that I can tell.

An exercise I'm running...  I'm asking students what happens to the speed necessary to maintain a circular orbit if (a) the central star's mass is doubled, (b) the planet's mass is doubled, or (c) the planet's distance from the central star is doubled.  I hand everyone a different half-page of paper with one of these three questions asked; for more variety, some of the papers say "tripled" or "quadrupled" rather than doubled.  Students are guided to solve in variables for the speed, then to use semi-quantitative reasoning to see what happens to the speed.

Then, I pull up "my solar system", a phet simulation.  Using the "sun and planet" preset, students are asked to change the simulation as described on their paper to see if they get a circular orbit.  (Those who told me that changing the satellite's mass changes its orbital speed as well become confused a bit when the simulation doesn't verify their answer.)


2 comments:

  1. This brings up another question I've been having about AP Physics 1. Since mathematical skill is de-emphasized, how important is it for students to know how to derive equations? For example, in AP Physics B, I made sure my students knew how to derive v and T for a satellite in a circular orbit, just in case a question like the one from the 2001 AP Physics C-Mech exam showed up on the AP Physics B exam. Even though I showed the derivation of these formulas in my current AP Physics 1 class, is it important for me to make sure they can derive them on a test?

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  2. I would not emphasize rote derivation of formulas for a test. I *would* make sure students understand where various formulas come from -- e.g. most of the orbital formulas come from Fnet=ma with Fnet given by Newton's law of gravitation, and a being v^2/r.

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