Final answer:
The total mechanical energy of a comet in an elliptical orbit around a star is negative. This aligns with the principles of orbital mechanics and Kepler's laws, which explain the dynamics of a comet's speed relative to its position in the orbit.
Step-by-step explanation:
The sign of the total mechanical energy of a comet in an elliptical orbit around a star, which moves counterclockwise, is negative. This is consistent with the principles of orbital mechanics, where an object in a bound elliptical orbit around another massive object, such as a star, has a negative total mechanical energy.
This unifies both the potential and kinetic energies the comet has in its orbit. The proof of why the total energy is negative, even for elliptical orbits, is tied to the conservation of energy and can be shown through calculations similar to those for circular orbits.
Kepler's second law of planetary motion, which states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time, explains why a comet travels much faster when it is close to the star (perihelion) and slower when it is farther away (aphelion). This law implies that as the comet gets closer to the star and the segment area remains constant, the comet must move faster to sweep out an equal area in an equal amount of time.