Kepler's Laws

Answer: D. Kepler's first law states that planetary orbits are ellipses with the Sun at one focus (NCERT Class 11 Physics Chapter 7, page 129).

Why A is wrong: A is wrong because Kepler corrected the circular-orbit assumption; orbits are elliptical, and the Sun is at a focus, not the geometric centre.

Why B is wrong: B is wrong because while the shape is an ellipse, the Sun occupies one focus — not the centre. The centre of an ellipse is equidistant from both foci.

Why C is wrong: C is wrong because a parabola describes an unbound trajectory (escape path), not a bound planetary orbit.

Answer: D. The gravitational force is central (directed along the radius vector), so torque about the Sun is zero and angular momentum is conserved. Equal areas in equal times follows from L = constant (NCERT Class 11 Physics Chapter 7, page 129).

Why A is wrong: A is wrong because the gravitational force continuously changes the planet's direction, so linear momentum is not conserved. It is angular momentum (about the Sun) that is conserved due to zero torque.

Why B is wrong: B is wrong because while total energy is conserved in the orbit, the equal-area law follows specifically from angular momentum conservation (zero torque), not from the energy equation.

Why C is wrong: C is wrong because mass conservation is trivially true for the planet but has no connection to the areal velocity being constant.

Answer: B. By conservation of angular momentum (Kepler's second law), m v₁ r₁ = m v₂ r₂, giving v₂ = v₁ r₁ / r₂ (NCERT Class 11 Physics Chapter 7, page 129).

Why A is wrong: A is wrong because it inverts the ratio. Since r₂ > r₁, this would make the aphelion speed greater than perihelion speed, contradicting Kepler's second law (planet moves slower when farther).

Why C is wrong: C is wrong because it uses an inverse-square ratio of distances. Angular momentum conservation gives a linear ratio (v ∝ 1/r for the transverse component), not an inverse-square one.

Why D is wrong: D is wrong because it applies the square of the distance ratio in the wrong direction, yielding a speed much larger at aphelion — the opposite of what Kepler's law requires.

Answer: C. By Kepler's third law, T² ∝ a³. If a₂ = 2a₁, then T₂² = 2³ T₁² = 8T₁², so T₂ = √8 T = 2√2 T (NCERT Class 11 Physics Chapter 7, page 129). (Trap: treating T as proportional to a gives the wrong answer 2T.)

Why A is wrong: A is wrong because it assumes T ∝ a (linear scaling). Kepler's third law gives T ∝ a^(3/2), so doubling a multiplies T by 2^(3/2) = 2√2, not 2. This is a common NEET distractor exploiting the linear-scaling confusion.

Why B is wrong: B is wrong because it assumes T ∝ a² (square scaling). Kepler's third law gives T² ∝ a³, i.e., T ∝ a^(3/2). Doubling a gives T₂ = 2^(3/2) T = 2√2 T ≈ 2.83T, not 4T.

Why D is wrong: D is wrong because it uses T ∝ a³. While T² ∝ a³ is correct, taking the square root gives T ∝ a^(3/2), so T₂ = 2^(3/2) T = 2√2 T, not 2³ T = 8T.

Answer: A. T ∝ a^(3/2). Factor = 4^(3/2) = (√4)³ = 2³ = 8 (NCERT Class 11 Physics Chapter 7, page 129). (Trap: linear scaling gives 4; square scaling gives 16.)

Why B is wrong: B is wrong. There is no standard power law that yields a factor of 6 from a 4× change in a. Kepler's law gives T ∝ a^(3/2), so the factor is 4^(3/2) = 8.

Why C is wrong: C is wrong because it assumes T scales linearly with a. Kepler's third law gives T ∝ a^(3/2), so quadrupling a multiplies T by 4^(3/2) = 8, not 4.

Why D is wrong: D is wrong because it assumes T ∝ a² (i.e., T factor = 4² = 16). The correct relation is T ∝ a^(3/2), giving 4^(3/2) = 8.

Answer: C. First derive how speed depends on radius: equating gravitational force to centripetal force, GMm/r² = mv²/r, gives v² = GM/r, so v ∝ 1/√r. Increasing r by 25× therefore decreases v by a factor of √25 = 5, i.e., v_new = v_old/5 — this is a separate result, not given in the stem. Second, feed that result into T = 2πr/v: T_new/T_old = (r_new/r_old) × (v_old/v_new) = 25 × 5 = 125. Neither step alone gives the answer; the velocity-scaling result from the force balance must be produced first and then substituted into the period relation (NCERT Class 11 Physics Chapter 7, page 129).

Why A is wrong: A is wrong because it applies only the √25 = 5 velocity-scaling factor on its own, forgetting that T = 2πr/v also carries an explicit factor of r itself; both the radius change and the resulting velocity change must enter T's scaling.

Why B is wrong: B is wrong because it applies only the radius factor (r × 25) to T, treating v as if it stayed constant — but the force-balance relation v ∝ 1/√r shows v also changes with r, and that change must be folded into T = 2πr/v.

Why D is wrong: D is wrong because it multiplies the radius factor by itself (25 × 25) instead of by the velocity factor (25 × 5); since v ∝ 1/√r gives v_new = v_old/5 (not v_old/25), the correct combination is 25 × 5 = 125, not 625.

Answer: C. Kepler's second law states equal areas are swept in equal time intervals, regardless of the planet's position in its orbit (NCERT Class 11 Physics Chapter 7, page 129). The areal velocity dA/dt is constant.

Why A is wrong: A is wrong because although the orbital radius is larger at aphelion, the planet's speed is correspondingly slower. Kepler's second law guarantees the area swept per unit time is the same everywhere in the orbit.

Why B is wrong: B is wrong for the same reason — the slower speed at aphelion exactly compensates for the larger radius, keeping dA/dt constant.

Why D is wrong: D is wrong because Kepler's second law holds for all elliptical orbits regardless of eccentricity. No additional information is needed.

Answer: B. T_Y/T_X = (a_Y/a_X)^(3/2) = (1/3)^(3/2) = 1/(3√3). So T_Y = 27/(3√3) = 27/(3 × 1.732) = 9/1.732 ≈ 5.20 years = 3√3 years (since 3√3 ≈ 5.196). Equivalently: (1/3)^(3/2) = 1/√27 = 1/(3√3), and 27 × 1/(3√3) = 27/(3√3) = 9/√3 = 3√3 (NCERT Class 11 Physics Chapter 7, page 129).

Why A is wrong: A is wrong because it assumes T ∝ a (linear): 27 × (1/3) = 9. Kepler's law gives T ∝ a^(3/2), so the factor is (1/3)^(3/2) = 1/(3√3), not 1/3.

Why C is wrong: C is wrong because it assumes T ∝ a² (or equivalently, divides by 3²= 9): 27/9 = 3. The correct exponent is 3/2, giving 27/(3√3) = 3√3 ≈ 5.2 years.

Why D is wrong: D is wrong because it assumes T ∝ a³: 27 × (1/3)³ = 1. That would mean T² ∝ a⁶, not T² ∝ a³. The correct result is 3√3 years.