Calculating the lenght of a year on mars?

[cruisx]

Homework Statement

The mean radius for the orbit of the planet Mars about the sun is 2.28 * 10¹¹m. If the radius of the orbit of the Earth around the sun is 1.49 * 10¹¹m. If it takes the Earth 1 year (365 days) to orbit the sun, how long (in Earth days) is the year on mars?

So this is the question is giving me trouble and was wondering if someone could help me out.

Homework Equations

4pi² * Ro / To² = Gme/ Ro²

I think this equation might be use full but i am not sure.

The Attempt at a Solution

I think i have the period of the Earth because 365 days = 31 556 926s correct?
Sorry i am not sure how much information i have to work with here and i am just stuck.

 

[jbsteele]

You can use the equation you wrote:T2 = 4pi2 * Ro / GmeWhere T is the period of the orbit, Ro is the radius of the orbit, and Gme is the gravitational constant times the mass of the Earth. Since you already know the radius of Mars' orbit and the period of Earth's orbit, you can calculate Gme. Then, you can use the same equation to calculate the period of Mars' orbit.

 

[tomcue]

The length of a year on Mars can be calculated using Kepler's Third Law, which states that the square of the orbital period (time it takes to complete one full orbit) is proportional to the cube of the semi-major axis (average distance from the planet to the sun).

Using the given information, we can set up the following equation:

(Tm)^2 / (Te)^2 = (Rm)^3 / (Re)^3

Where Tm is the orbital period of Mars, Te is the orbital period of Earth, Rm is the radius of Mars' orbit, and Re is the radius of Earth's orbit.

Substituting in the values given, we get:

(Tm)^2 / (365 days)^2 = (2.28 * 10^11 m)^3 / (1.49 * 10^11 m)^3

Solving for Tm, we get:

Tm = (365 days) * sqrt((2.28 * 10^11 m)^3 / (1.49 * 10^11 m)^3)

Tm = 687 days

Therefore, a year on Mars is approximately 687 Earth days long.