Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orbits (1 Viewer)

[frog0101]

Hi,
For the dotpoint:
Investigate the relationship of Kepler’s Laws of Planetary Motion to the forces acting on, and the total energy of, planets in circular and non-circular orbits.
What are we meant to do (or use) for the underlined part (This is almost the exact same as in next DP). Does anyone have any notes or information for this?

Also, are the equations (given with that dotpoint) restricted to circular orbits (I believe, accurately, they only apply for circular) for the course.

Thanks

 

[weaknuclearforce]

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb

Hey frog0101, I'm sorry I have no idea what the hsc wants in regards to the phrase "investigate", perhaps a qualitative explanation?

Quantitatively, circular / elliptical trajectories can be derived using Euler Lagrange equations where the Lagrangian (in polar co-ordinates) would be

Using conserved angular momentum and plugging into energy (Hamiltonian) = T + U this can all be massaged into an equation which has a trajectory form with an epsilon parameter

If epsilon = 0 you have a circular trajectory, if it is between 0 and 1 then elliptical - you can use a grapher app to easily check this.
Your epsilon parameter will contain Energy, angular momentum squared, mass and coefficient's of a Newtonian potential squared, so it is nice to note that only Energy can be negative in that lump! To get to this point requires some effort! It begins with calculus of variations, needs knowledge of polar co-ords and then a not so trivial integral. If any of this is what you need I recommend following Duric's excellent Astrophysics book where all these steps are laid out in chapter 1.
Perhaps the "investigate" term means to acknowledge and quote these facts and then describe the trajectories based on how Energy affects the value of epsilon? I'm betting advanced mechanics does require you to know of Lagrangian's and the EL equations, so you could at least blast through that early math? It's worth doing.

 

[Arrowshaft]

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb

Lol, although it's called Advanced Mechanics, our course is strictly NON calculus. Even derivations requiring calculus (for the most elegant proofs) such as the derivation of the formula for centripetal force only uses small angle approximations. It's weird, new syllabus brings some topics of motion from the 4u syllabus such as angular velocity, displacement and circular motion, yet they decide to drop the calculus.

 

[Arrowshaft]

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb

Probably because they can't assume all physics students are taking at least 2u math. While they most definitely should, they cannot make that assumption unless they require that all physics students do so, or if they teach the calculus in physics (which is not going to happen since it's a physics course, not a math course).

Yeah I know, which is why i wish they had two different physics courses, a calc course and a non-calc.


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[InteGrand]

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb

Why are they calling it 'Advanced' Mechanics if there won't be any calculus? Do they mean 'advanced' as in relative to the Moving About topic (previous syllabus)?

 

[weaknuclearforce]

Re: Kepler's Laws of Planetary Motion - Total energy of Circular and non-circular orb

My apologies, I thought maybe it might have been a 'research yourself' type investigation/project similar to the extension science unit. It's certainly feasible to quote results (such as the integral I mentioned) and then discuss the implications in terms of energy.