Question 366: Gravitation

Question

The radius of Martian orbit around the Sun is about 4 times the radius of the orbit of Mercury. The Martian year is 687 Earth days. Then which of the following is the length of 1 year on Mercury?

Accepted Answer

To determine the length of one year on Mercury, we use Kepler's Third Law, which states that the square of the orbital period $ T $ is proportional to the cube of the mean radius $ R $ of the orbit:
$ T^2 \propto R^3 $
Given:
The radius of Mars' orbit around the Sun, $ R' = 4R $, where $ R $ is the radius of Mercury's orbit.
The Martian year $ T' = 687 $ Earth days.
First, apply Kepler's Third Law to find the ratio of the orbital periods:
$ \left(\frac{T'}{T}ight)^2 = \left(\frac{R'}{R}ight)^3 = \left(\frac{4R}{R}ight)^3 = 4^3 = 64 $
From this, we can solve for the ratio of the periods:
$ \frac{T'}{T} = 8 $
This means that Mars takes 8 times longer to orbit the Sun compared to Mercury. Therefore, the length of one year on Mercury is:
$ T = \frac{T'}{8} = \frac{687}{8} \approx 85.88 	ext{ days} $
Hence, one year on Mercury is approximately 86 Earth days.

Answer

172 earth days

Answer

124 earth days

Answer

88 earth days

Answer

225 earth days

Gravitation

The radius of Martian orbit around the Sun is about 4 times the radius of the orbit of Mercury. The Martian year is 687 Earth days. Then which of the following is the length of 1 year on Mercury?

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