Preliminary Calculations by NASA

A satellite is a thing that revolves around a massive body of mass. Satellites built by humans are sent into orbit for a variety of uses, including communication, scientific research, weather forecasting, and intelligence. A projectile is the movement of a satellite around a large body. The only force acting on the satellite once it is in space is gravity. However, any satellite will orbit any massive body if it is launched quickly enough. When launching satellites to orbit around the Mars, scientists have to determine the exact speed to launch the satellite so that it can remain on the orbit (Yvelyne 86). From the laws governing a projectile launched body, a satellite moves in a direction that is tangent to the Mars. As a result, the force of gravity of the Mars acts to pull it down. If the commence speed is too small, the launched body would fall to the Mars as the Mars’s gravitational forces would pull it down. When started with sufficient speed, the body would take a circular path and would fall to surface or Mars (Léonie, Léonie and Grady 13). When the launch speeds are made to be significantly large, the projected body moves at an elliptical path. At every point along the trajectory path, the satellite falls towards the surface of the Mars. However, it does not reach the surface of the Mars. For the case of this essay, the primary objective is to develop preliminary calculations. The scheming will show how high the satellite must be placed above the surface of the planet Mars and the speed that it must maintain while on the orbit.

Laws

The laws governing the motion of a satellite around the mars will be Newton’s laws. The law of acceleration and Newton’s form of Kepler’s third law will be applicable in determining the relationship between the gravitational force acting on the planet and the period.

Calculations

For instance, considering the mass of planet Mars to be and the mars of the satellite to be, it is possible to customize Newton’s gravitational laws to one that can be used for this problem. Planet Mars can be the central object while the satellite will revolve around the planet (Stephanie 75). Therefore, planet Mars will cause a sufficient acceleration on the satellite. Assuming that the satellite moves in a circular motion, centripetal forces will act upon the orbiting satellite and will be given by the relationship:

This net force arises from the gravitational force that attracts the satellite towards the central body. The force of gravity will have the value:

Since the force of gravity is equivalent to the net centripetal force, it is possible to combine the two equations to get:

Since the mass of the satellite is present in both formulas, it cancels out to give rise to:

The term R will also simplify to obtain:

The above formula can be used to determine the speed required to launch the satellite so that it can revolve around Mars without falling.

The above equation shows that the force of gravity is the only force that will determine the speed with which the satellite will revolve around the planet Mars. The gravitational force of planet Mars has the value. The mass of planet Mars is (Stephanie, 34).

The gravitational law is not enough to explain the case of a satellite moving around mars. Other laws such as acceleration and Kepler’s third law will be very applicable to this case. The mass and radius of planet Mars will help to determine the acceleration of the satellite. Since the only force acting on the moving satellite is the gravitational force, the magnitude of the acceleration will be equivalent to the acceleration by the gravity of the satellite at any particular location or instance ion location. Therefore:

The term R represents the radius of the orbit for the satellite.

The Newton’s form of Kepler’s third law also describes the motion of a satellite around Mars. The satellite will move around the entire circumference given by the formula:

The period of the satellite will be given by the formula:

The equation representing the period of the satellite, and the mean distance from the center of the body (for this case the body is Mars), the equation will take the form:

The equation above gives the period of the satellite as it moves around the planet.

Results

For us to get an effective solution for the case of this satellite, we assume that the satellite will be stationary. The main reason for making this assumption is because stationary satellites are the most common forms used on earth. A stationary object is one that moves with the speed of the planet and has a period that is equal to one rotation of the planet. Mars makes one complete rotation after. There, the period for this planet will be:

Obtaining the speed of the satellite requires the radius of the orbit. From the equation of the period, it is possible to obtain the radius of the orbit.

Particularly, this value gives the distance that the satellite will be from the center of the mars. Therefore, it is possible to determine the exact location of the satellite from the surface of Mars. Planet Mars has a radius of size. Therefore, the exact distance of the satellite from the surface of Mars is , where d is the distance from the surface of Mars to the satellite and r is the radius of Mars.

The above results show that the radius of Mars is significantly small when compared to the distance of the satellite from the surface of Mars. The velocity that the satellite must be launched at will have the value:

Conclusion

The calculations carried out have revealed the possibility of determining the exact location of a satellite, as well as the speed with which the satellite must be launched with. However, the results are based on assumptions that the satellite was stationary. In case, the satellite has a different period; it is possible to use the developed formulations to determine the launch speed and the exact location. The results obtained are very effective when launching a satellite since it will be easy to launch it at the right speed that will ensure it orbits around Mars.

Works Cited

Léonie, J. Rennie, et al. Integrating Science, Technology, Engineering, and Mathematics: Issues, Reflections, and Ways Forward. Boston: Routledge, 2012. Pdf.

Stephanie, Paris. Leveled Texts for Mathematics: Data Analysis and Probability. New York: Teacher Created Materials, 2011.

-. Leveled Texts for Mathematics: Data Analysis and Probability. New York: Teacher Created Materials, 2011.

Yvelyne, Germain-McCarthy. Mathematics and Multi-Ethnic Students: Exemplary Practices. London: Taylor & Francis, 2017.