Question

A planet is moving in an elliptical orbit. If T, U, E and L are its kinetic energy, potential energy, total energy and magnitude of angular momentum respectively, then which of the following statements is true.
A) T is conserved
B) U is always positive
C) E is always negative
D) L is conserved but the direction of vector $\overrightarrow L $ will continuously change

Answer 1

Hint: The planet is moving in an elliptical orbit. The orbit is elliptical so the distance between center and the planet will change and so does velocity of the planet. The motion of the planet will be planar. Find the direction of angular momentum by finding the direction of displacement of the planet.

Complete solution:
Let's discuss whether each of the three energies and momentum remains constant or change throughout the motion.
Kinetic energy (T): The path of the planet is given to be elliptical, so the distance between the center and the planet will change. Due to this change, the velocity of the planet will change. As a result, the kinetic energy will not be conserved.
Therefore, option A is incorrect.

Potential energy (U): For any attractive field, the potential energy is negative. Since the work is done against the acceleration, hence there is always a negative sign potential energy.
Therefore, option B is also incorrect.

Angular momentum: Since no external torque is acting on the system, hence the angular momentum of the system is conserved. But for an elliptical orbit which is a plane, the direction of angular momentum will not change.
Hence option D is also incorrect

Total energy: An object with negative total energy is gravitationally bound and therefore is in orbit. Thus, E must be negative.

Hence, Option C is the correct answer.

Note: Remember if an object in orbit has positive total energy then the object will deflect from the orbit. The direction of angular momentum does not change for an orbital motion. The kinetic energy keeps changing due to change in velocity.