Understanding Kepler's Laws of Planetary Motion

Kepler's laws describe the motion of planets around the Sun. There are three main laws. Let's examine the given statements in the context of these laws.

Analyzing Statement (i): Speed Variation

Statement (i) says: "Planets move faster at perihelion and slower at aphelion."

• Perihelion is the point in a planet's orbit where it is closest to the Sun.
• Aphelion is the point where it is farthest from the Sun.

This statement relates to Kepler's Second Law, which is about the conservation of angular momentum. For a planet's orbit, angular momentum is conserved. This means that as the distance from the Sun decreases (moving towards perihelion), the planet's speed must increase to keep the angular momentum constant. Conversely, as the distance increases (moving towards aphelion), the speed must decrease.

Therefore, statement (i) is correct.

Analyzing Statement (ii): Equal Areas in Equal Times

Statement (ii) says: "A line joining a planet to the sun sweeps out equal areas in equal intervals of time."

This is the precise mathematical statement of Kepler's Second Law of Planetary Motion. It directly follows from the conservation of angular momentum. Imagine a line connecting the planet to the Sun; over any given time period (e.g., one hour), the area this line 'draws' or 'sweeps' through space is constant, regardless of whether the planet is near or far from the Sun.

Therefore, statement (ii) is correct.

Analyzing Statement (iii): Speed at Perihelion and Aphelion

Statement (iii) says: "Planets moves slower at perihelion and faster at aphelion."

As explained when analyzing statement (i), this contradicts the principle derived from Kepler's Second Law. Planets move faster when they are closer (perihelion) and slower when they are farther away (aphelion). This statement is the exact opposite of what happens.

Therefore, statement (iii) is incorrect.

Conclusion on Valid Statements

Based on the analysis:

• Statement (i) is valid.
• Statement (ii) is valid.
• Statement (iii) is invalid.

Thus, the valid statements are (i) and (ii) only.