Question at position 27 When a planet, in its orbit, is closer to the Sun, it:moves slower than averagefeels less gravitational pull than averagemoves faster than averagereflects less sunlight than averagespins faster on its axisClear my selection

Question

BlackTom AI · Accepted Answer

Kepler's second law (equal areas in equal times) and the vis-viva equation govern orbital speed: a planet sweeps out equal areas in equal times, so it must move faster when it is nearer the Sun.

1) Relevant concepts
- Kepler’s second law: line joining planet and Sun sweeps equal areas in equal times, implying variable orbital speed.
- Gravitational force law: F = GMm/r^2 (stronger when r is smaller).
- Vis‑viva (energy) equation for instantaneous orbital speed: v^2 = GM(2/r − 1/a), where r is current distance, a is semi-major axis, GM is Sun’s gravitational parameter.

2) Step-by-step reasoning
- From Kepler’s 2nd law: area element dA = 1/2 r^2 dθ, so for fixed dA/dt a smaller r requires a larger dθ/dt. Linear speed v = r dθ/dt, so decreasing r increases dθ/dt enough that v increases.
- From vis‑viva: v^2 = GM(2/r − 1/a). For fixed a, as r decreases the term 2/r grows, so (2/r − 1/a) increases → v^2 increases → v increases.

3) Calculations (intermediate steps)
Take a representative ellipse with a = 1 AU and GM normalized = 1 for simplicity.
- At r = 0.7 AU: v^2 = 1*(2/0.7 − 1/1) = 2.8571 − 1 = 1.8571 → v ≈ 1.363 (units).
- At r = 1.3 AU: v^2 = 2/1.3 − 1 = 1.5385 − 1 = 0.5385 → v ≈ 0.734.
This numerical example shows the planet is substantially faster when closer to the Sun.

4) Why the correct answer is correct
Both Kepler’s area law and the vis‑viva equation show that a smaller r produces a larger orbital speed. Thus “moves faster than average” correctly describes the motion near perihelion.

5) Why other options are incorrect
- “Moves slower than average”: contradicted by Kepler’s 2nd law and vis‑viva.
- “Feels less gravitational pull than average”: wrong because F ∝ 1/r^2, so gravity is stronger when closer.
- “Reflects less sunlight than average”: solar flux ∝ 1/r^2, so closer means more—not less—sunlight per unit area.
- “Spins faster on its axis”: orbital distance does not generally change a planet’s axial spin rate; there is no direct causal link for ordinary planetary orbits.

6) Conclusion
Therefore the correct choice is “moves faster than average,” which matches the provided answer.

Question at position 27 When a planet, in its orbit, is closer to the Sun, it:moves slower than averagefeels less gravitational pull than averagemoves faster than averagereflects less sunlight than averagespins faster on its axisClear my selection单项选择题

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