Question

The kinetic energies of a planet in an elliptical orbit about the Sun, at positions A, B and C are K_A, K_B and K_C, respectively. AC is the major axis and SB is perpendicular to AC at the position of the Sun S as shown in the figure. Then

• A. K_B > K_A > K_C
• B. K_A < K_B < K_C
• C. K_B < K_A < K_C
• D. K_A > K_B > K_C

Hint:

According to Kepler’s 2nd Law of planetary motion, a line segment joining the sun and planet sweeps out equal areas during equal intervals of time.

Answer 1

The Correct Option is D

To solve this problem, we need to consider the properties of elliptical orbits, specifically that of a planet orbiting the Sun. According to Kepler's laws and the conservation of angular momentum, the kinetic energy of a planet in an elliptical orbit varies with its position.

Concepts Involved:

• Kepler's Second Law: This law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies the planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther from the Sun (aphelion).
• Kinetic Energy and Orbital Speed: The kinetic energy of the planet is given by the equation K = \frac{1}{2}mv^2, where m is the mass of the planet and v is its orbital speed. Faster velocity indicates higher kinetic energy.

Analysis of Positions A, B, and C:

1. Position A: This position is closer to the Sun than position C, but not the closest point.
2. Position B: Position B is perpendicular to the major axis AC at the Sun, indicating it is not at the extreme of the orbit, but closer to the Sun compared to position C. Thus, the planet should have a relatively high velocity here.
3. Position C: This position is at the extreme end of the major axis, away from the Sun (aphelion). Thus, the velocity and kinetic energy are expected to be the lowest here.

Conclusion:

Based on the above analysis, the relative order of kinetic energies should be:

• Position A (closer to perihelion) has higher kinetic energy than position B. Hence, K_A \gt K_B.
• Position B (closer than C but not on the major axis end) has higher kinetic energy than position C. Hence, K_B \gt K_C.

Therefore, the correct order is K_A \gt K_B \gt K_C.