Question

the table gives the orbital radius of four planets.
planet orbital radius

planet	orbital radius (× 1,000,000 km)
mars	228
mercury	58
venus	108

according to the table, which statement accurately compares the length of one year for two of these planets?
a earth has a longer year than mars.
b mars has a longer year than venus.
c venus has a longer year than earth.
d mercury has a longer year than venus.

Explanation:

To solve this, we use Kepler's third law, which states that the square of the orbital period (length of a year) is proportional to the cube of the orbital radius. So, a larger orbital radius means a longer orbital period (longer year).

Step 1: Recall Kepler's Third Law

Kepler's third law: $T^2 \propto r^3$, where $T$ is the orbital period (year length) and $r$ is the orbital radius. Thus, larger $r$ implies larger $T$.

Step 2: Compare Orbital Radii

• Earth: 150, Mars: 228. Since 150 < 228, Mars has a larger radius, so Mars should have a longer year than Earth. So option A is wrong.
• Mars: 228, Venus: 108. 228 > 108, so Mars has a larger radius. Thus, Mars has a longer year than Venus. Let's check other options to confirm.
• Venus: 108, Earth: 150. 108 < 150, so Earth has a larger radius, meaning Earth has a longer year than Venus. So option C is wrong.
• Mercury: 58, Venus: 108. 58 < 108, so Venus has a larger radius, meaning Venus has a longer year than Mercury. So option D is wrong.

Answer:

B. Mars has a longer year than Venus.