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Physics Question 31 – JEE-MAIN 2025

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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant. Reason (R) : For a central force field the angular momentum is a constant. In the light of the above statements, choose the most appropriate answer from the options given below :

Recall the statement of Kepler's Second Law regarding the areal velocity of a planet.

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Ninja Strategy•Fundamental Principle Check

Recognize that both Kepler's Second Law (A) and the conservation of angular momentum in a central force field (R) are fundamental and correct principles, then determine if R explains A.

Step 1: Evaluate Assertion (A)✦ Active

Assertion (A) states that the radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time, implying constant areal velocity. This is precisely Kepler's Second Law of planetary motion. Therefore, Assertion (A) is correct.

Step 2: Evaluate Reason (R)○ Expand

Reason (R) states that for a central force field, the angular momentum is constant. The gravitational force between the Sun and a planet is a central force. For a central force, the torque τ→=r→×F→ about the center of force is zero because the force vector F→ is always directed along the position vector r→ (or opposite to it). Since torque is the rate of change of angular momentum (τ→=dL→/dt), a zero torque implies that the angular momentum L→ is conserved (constant). Therefore, Reason (R) is correct.

Step 3: Determine Relationship between (A) and (R)○ Expand

Kepler's Second Law (constant areal velocity) is a direct consequence of the conservation of angular momentum. The areal velocity dA/dt is related to the angular momentum L by the formula dA/dt=L2m, where m is the mass of the planet. Since Reason (R) correctly states that angular momentum L is constant for a central force field (like gravity), it directly explains why the areal velocity dA/dt is constant, as stated in Assertion (A). Thus, Reason (R) is the correct explanation for Assertion (A).

💡 Teacher's Secret Hint

Remember that Kepler's laws are empirical laws derived from observations, but they can be theoretically derived from Newton's laws of motion and gravitation, with conservation of angular momentum being a key principle.

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