the spacecraft is first put into a circular orbit around Mars. The period of the orbit is 110 minutes. Calculate the altitude of the orbit in km.
To calculate the altitude of the orbit around Mars, we can use Kepler's third law of planetary motion, which states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of the orbit.
The formula is as follows:
T^2 = (4π^2 / GM) * a^3
Where:
T = Orbital period of the spacecraft (in seconds)
G = Gravitational constant (approximately 6.67430 x 10^-11 N*m^2/kg^2)
M = Mass of Mars (approximately 6.39 x 10^23 kg)
a = Semi-major axis of the orbit (in meters)
In the given problem, the orbital period is 110 minutes, which is equal to 6600 seconds.
Now, let's rearrange the formula to solve for the altitude (a):
a^3 = (T^2 * GM) / (4π^2)
a = cube root of [(T^2 * GM) / (4π^2)]
First, we need to convert the orbital period from minutes to seconds:
T = 6600 seconds
Now, we can substitute the known values into the formula:
a = cube root of [(6600^2 * 6.67430 x 10^-11 * 6.39 x 10^23) / (4π^2)]
Evaluating this expression will give us the semi-major axis of the orbit. To convert the semi-major axis from meters to kilometers, divide the result by 1000.
By following these steps, you should be able to calculate the altitude of the spacecraft orbiting Mars.