Question

How do you calculate the altitude and velocity of a satellite in a geosynchronous orbit of mars?

Answer 1

Hint: In this question we have to explain the method to calculate the altitude and velocity of a satellite in a geosynchronous orbit of mars. So, first we will understand what a geosynchronous orbit is. Then we will understand the method to find the altitude and velocity of a satellite in a geosynchronous orbit of mars.

Complete step by step answer:
Geosynchronous orbit- A geosynchronous orbit is a high earth orbit that allows satellites to match the rotation of earth. This orbit is located at 35,786 km above earth's equator.
To find geosynchronous orbit of any planet, we must take following steps:
First we must know the centripetal force exerted on the object in circular motion. Using following formula,
${F_c} = \dfrac{{m{v^2}}}{r}$
Then we will find the gravitational force of attraction between two objects. Using following formula,
${F_c} = \dfrac{{GMm}}{{{r^2}}}$
Where,
M is the mass of mars and m is the mass of satellite
r is the distance between them
G is the gravitational constant
Now, for the satellite to be stable in the orbit, the centripetal force and the gravitational force must be equal to each other.
$\dfrac{{m{v^2}}}{r} = \dfrac{{GMm}}{{{r^2}}}$
By simplifying above equation we get following relation,
$v = \sqrt {\dfrac{{GM}}{r}} $
If the time period of rotation is T and the orbit length is$2\pi r$. So, we can write following equation;
$v = \dfrac{{2\pi r}}{T}$
It implies following equation;
$\dfrac{{2\pi r}}{T} = \sqrt {\dfrac{{GM}}{r}} $
By simplifying this we get following equation;
$r = \sqrt[3]{{\dfrac{{GM{T^2}}}{{4{\pi ^2}}}}}$
By using the above relation we can find the distance. This distance is the distance from the center of mars. To find altitude we will subtract the radius of mass from r.
To find the velocity we will use following formula;
$v = \dfrac{{2\pi r}}{T}$
By using the above formula we can find the velocity of the satellite in the geosynchronous orbit of mars.

Hence, from the above explanation we can clearly understand the method to calculate the altitude and velocity of a satellite in a geosynchronous orbit of mars.

Note: It is clear that finding the altitude and velocity for the geosynchronous orbit of mars is a tedious task. Same method can be used to find the altitude and velocity of different plants.