JEE Main 2025 · 7 April · evening · Single correct · +4 / −1
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) (A) is not correct but (R) is correct
- (b) Both (A) and (R) are correct but (R) is not the correct explanation of (A)
- (c) Both (A) and (R) are correct and (R) is the correct explanation of (A)
- (d) (A) is correct but (R) is not correct
Answer and solution
Answer (c) Both (A) and (R) are correct and (R) is the correct explanation of (A)
Solution
Physics sir · 4 steps
Gravity pulls along the Sun–planet line, so it has no torque about the Sun, and that fixes the areal velocity.
- Central force. The Sun's pull on a planet always points along the line joining them, so F⃗ is parallel to r⃗. The torque about the Sun is τ⃗ = r⃗ × F⃗ = 0, so the angular momentum L⃗ stays constant. (R) is correct.
- Area swept. In a short time dt the radius vector sweeps a thin triangle of area dA = 1/2|r⃗ × v⃗ dt|. Since L⃗ = m r⃗ × v⃗, this ties the areal velocity directly to L.
dA/dt = 1/2|r⃗ × v⃗| = L/2m
- Link them. With L and m both constant, dA/dt is constant, so equal areas are swept in equal times. That is (A), Kepler's second law, and it follows straight from (R).
- So. Both statements are true and conservation of angular momentum under a central force is exactly why the areal velocity is constant. The answer is (c).
The idea to keep. Kepler's second law is angular momentum conservation in disguise: any central force gives zero torque, constant L, and so constant areal velocity L/2m.
Answer from the NTA final answer key. Solution by PrepWiser.