Mechanical energy for planet in elliptical orbit around star

[JessicaHelena]

Homework Statement

A planet is in an elliptical orbit around a star. Which of the following best represents the mechanical energy E_planet of just the planet and the mechanical energy Es_tar-planet of the star-planet system as functions of time for one complete orbit?

Homework Equations

Ei = Ef (?)

The Attempt at a Solution

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The planet has both PE and KE, and when it is further away from the star, it is traveling much faster so KE goes up while PE goes down due to increased r. And when it's near the star, r increased so PE goes up but KE decreases due to slower speed. But the sum of PE and KE should be equal anytime.

For star-planet, the PE is -GmM/r, so I chose the answer A, but apparently the answer is C and I'm having a hard time understanding why.

 

[jbriggs444]

For star-planet, the PE is -GmM/r, so I chose the answer A, but apparently the answer is C and I'm having a hard time understanding why.

Like you, I would have chosen answer A, including both the planet's kinetic and gravitational potential energy in the planet's "total mechanical energy".

However, it appears that "mechanical energy of just the planet" is intended not to include any gravitational potential energy based on the planet's position in the gravitational field of the star.

The "mechanical energy of the star-planet system" does include the gravitational potential energy (-GmM/r) just as you indicate along with their kinetic energies. The total is constant and negative, as indicated by both A and C.

 

[JessicaHelena]

@jbriggs444 — then when the problem says "ME of just the planet", should I normally think of simply the KE? Does these kinds of things apply to other situations?

 

[jbriggs444]

@jbriggs444 — then when the problem says "ME of just the planet", should I normally think of simply the KE? Does these kinds of things apply to other situations?

In my view, the phrasing was ambiguous. So there is no hard and fast rule to cling to.

A useful clue might have been the fact that in answer A, the total mechanical energy of the planet was indicated as constant and positive. But with the conventional choice of zero potential energy at infinity, the energy should have been constant and negative.

That is a useful general approach to successful multiple choice problem solving -- if a particular interpretation of a problem makes all of the answers wrong, try a different interpretation.