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MCQEasyJEE 2025Kepler's Laws of Planetary Motion

JEE Physics 2025 Question with Solution

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 :

  • A

    Both (A) and (R) are correct and (R) is the correct explanation of (A)

  • B

    Both (A) and (R) are correct but (R) is not the correct explanation of (A)

  • C

    (A) is correct but (R) is not correct

  • D

    (A) is not correct but (R) is correct

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Assertion (A) states that the radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time, so the areal velocity is constant.

Find: Whether Assertion (A) and Reason (R) are correct, and whether Reason (R) explains Assertion (A).

Assertion (A) is Kepler's second law of planetary motion. It states that a line joining a planet and the Sun sweeps out equal areas in equal times. Therefore, the areal velocity is constant. So Assertion (A) is correct.

Reason (R) states that for a central force field, angular momentum remains constant. Gravitational force is a central force, and hence the torque about the Sun is zero. Therefore angular momentum is conserved. So Reason (R) is also correct.

The areal velocity is related to angular momentum by

dAdt=L2m\frac{dA}{dt} = \frac{L}{2m}

Since LL is constant in a central force field and mm is constant, dAdt\frac{dA}{dt} is constant.

Thus, Reason (R) correctly explains Assertion (A). Therefore, the correct option is A.

Concept Link

Given: Planetary motion under gravitational force.

Find: The correct assertion-reason relationship.

  1. Check Assertion (A): The statement about sweeping equal areas in equal intervals of time is exactly Kepler's second law. Hence, Assertion (A) is true.

  2. Check Reason (R): In a central force field, force acts along the line joining the two bodies. Therefore, torque about the center is zero, so angular momentum remains constant. Hence, Reason (R) is true.

  3. Check explanation: Constant areal velocity follows from conservation of angular momentum because

dAdt=L2m\frac{dA}{dt} = \frac{L}{2m}

So if angular momentum is constant, the areal velocity is also constant.

Therefore, both statements are correct and Reason (R) is the correct explanation of Assertion (A). The correct option is A.

Common mistakes

  • Students often treat Kepler's second law as an isolated fact and miss that it follows from a central force. This is wrong because the causal link is conservation of angular momentum. Always check whether the Reason explains the Assertion physically.

  • A common mistake is to think that angular momentum is conserved for every force. This is incorrect because angular momentum is conserved only when the torque about the chosen point is zero. For planetary motion, gravity is central, so the torque about the Sun is zero.

  • Some students confuse constant speed with constant areal velocity. These are not the same. In elliptical motion, the planet's speed changes, but the area swept per unit time remains constant.

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