# Calculus in the velocity and acceleration of satellites.

• VinnyO
This is calculus.In summary, the student was working on a project involving the velocity and acceleration of satellites and was recommended to include calculus. They attempted to use the derivative of the orbital speed equation to find acceleration, but realized their mistake when taking into account that velocity and acceleration are vectors. They then attempted to use integration to find velocity, but their calculations did not match the accepted acceleration equation.
VinnyO

## Homework Statement

I am working on a project dealing with the velocity and acceleration of satellites based on their distance from Earth. I was recommended to include some calculus in this project. Originally I thought I could just take the derivative of the orbital speed equation to find acceleration, given the mass of the Earth and the gravitational constant would be constants in the situation. When I tested this hypothesis with the accepted acceleration equation the answers came out different. I am fairly sure this is because the remaining variable is R (the radius of the orbit) as opposed to time, but perhaps I am making a different mistake. So at this point I am in search for a way to make calculus relevant in this situation.

v = SQRT(G*M/R)
a = G*M/R^2

## The Attempt at a Solution

I also have been looking for a way to calculate the distance of objects like the ISS from Earth based on observations/ measurements I can take myself (from Earth), but I assume any solution to this would be related to trigonometry or geometry.

VinnyO said:
take the derivative of the orbital speed equation to find acceleration
Remember that velocity and acceleration are vectors. Differentiating a speed (a scalar) does not give acceleration.

VinnyO
Remember that acceleration is change in velocity with respect to change in time

Moved to Calculus & Beyond section

haruspex said:
Remember that velocity and acceleration are vectors. Differentiating a speed (a scalar) does not give acceleration.
haruspex said:
Remember that velocity and acceleration are vectors. Differentiating a speed (a scalar) does not give acceleration.
Currently I have only taken one year of calculus in which I have not learned vectors (the report is for math in the program I am in and because I have taken calculus I was encouraged to utilize that skill). However with a bit of research on the basics of such I realize how obvious my mistake is. With that in mind my first though was that perhaps I could find velocity given the acceleration formula with some integration. But anyways in short this was my math:
I took an online example where R (the radius of the orbit) was 6.47 x 10^6 m. When plugged into the acceleration formula, a = (6.673 x 10^-11 N m^2/kg^2) • (5.98 x 10^24 kg) / (6.47 x 106 m)^2, a (acceleration) was equal to 9.53 meters per second squared.
To test my idea to use calculus I took the orbital speed equation, v = SQRT(G*M/R), and plugged in the gravitational constant, 6.673 x 10-11 N m^2/kg^2, for G and the mass of Earth, 5.98 x 10^24 kg, for M
I than simplified this to v=998806037 / 50*R^1/2
The derivative of this was a= -998806037 / 100R^3/2
I finally just plugged in the same R value into the new potential acceleration equation, a= -998806037 / 100(6.47x10^6)^3/2, and got a=6.07x10^-4 which was not equal to the answer produced by the equation so I knew I made some wrong assumption.

VinnyO said:
The derivative of this was a= -998806037 / 100R^3/2
That is the derivative wrt radius, dv/dR. Acceleration is the derivative of velocity wrt time, dv/dt.

## 1. How is calculus used in the study of satellite velocity and acceleration?

Calculus is used in the study of satellite velocity and acceleration to analyze the changes in these quantities over time. By using calculus concepts such as derivatives and integrals, scientists can calculate the instantaneous velocity and acceleration of a satellite at any given point in its orbit.

## 2. What is the relationship between satellite velocity and acceleration?

Satellite velocity and acceleration are related through the fundamental principle of calculus known as the derivative. The acceleration is the rate of change of velocity, meaning that the derivative of the velocity function gives the acceleration function.

## 3. How do scientists use calculus to predict the trajectory of satellites?

Scientists use calculus to predict the trajectory of satellites by analyzing the velocity and acceleration of the satellite at any given point in time. By integrating the acceleration function, they can find the displacement function, which gives the position of the satellite at any given time. This allows them to predict the path that the satellite will follow.

## 4. Can calculus be used to optimize satellite orbits?

Yes, calculus can be used to optimize satellite orbits. By analyzing the velocity and acceleration of a satellite, scientists can determine the most efficient path for the satellite to take in order to achieve its desired orbit. This involves finding the optimal combination of velocity and acceleration to minimize fuel consumption and maximize the satellite's efficiency.

## 5. What are some real-world applications of calculus in satellite technology?

Calculus is used in a variety of real-world applications in satellite technology, such as orbit determination, orbit maintenance, and orbit transfer. It is also used in the design and control of satellite systems, including calculating the necessary thrust and trajectory adjustments to keep satellites in their desired orbits.

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