# A question about a moving satellite (Kepler)

• sapz
In summary, the conversation discusses the motion of a satellite in an elliptic path with given initial conditions and equations. The satellite starts on the x axis and the question is how to find the z and y coordinates when x=0. The answer is that the initial conditions already provide the coordinates, as the satellite starts at the origin and the equation of the ellipse can be used to determine its path.
sapz

## Homework Statement

Hi

I have a question about a satellite moving in an elliptic motion. It begins its motion on the x axis, when x=2Re, and with a velocity: Vx=0, Vy=v0, Vz=-0.5V0.

Re is radius of the earth.

Solving this I arrived at:
r0 = 10Re/4.
e=1/4.
Rmin = 2Re.
Rmax = 10Re/3.

Given all that, How do I find the z and y coordinates of the satellite, when x=0?

## The Attempt at a Solution

sapz said:
Given all that, How do I find the z and y coordinates of the satellite, when x=0?
If you mean the initial conditions, you are given them: the object starts on the x axis, so y and z start off as zero.

No, I mean I want to find, during the elliptical motion, where the satellite will meet the plane ZY.

The origin is one focus, yes? Can you write down the equation of the ellipse?

To answer this question, we first need to understand the motion of the satellite in an elliptic orbit. Kepler's laws of planetary motion state that a satellite in an elliptic orbit around a central body will sweep out equal areas in equal times, and the square of the orbital period is proportional to the cube of the semi-major axis of the ellipse.

Using this information, we can calculate the orbital period of the satellite using the formula T^2 = (4π^2/GM) * a^3, where T is the orbital period, G is the gravitational constant, M is the mass of the central body (in this case, the Earth), and a is the semi-major axis of the ellipse.

Once we have the orbital period, we can calculate the angular velocity of the satellite using the formula ω = 2π/T. This will give us the rate at which the satellite moves along its elliptical orbit.

To find the z and y coordinates of the satellite when x=0, we can use the parametric equations for an ellipse: x = a*cos(ωt), y = b*sin(ωt), z = 0, where a is the semi-major axis and b is the semi-minor axis. We can plug in our known values for a and ω, and solve for y and z at the given time, t.

Alternatively, we can also use the formula for the position vector of an object in circular motion, r = a*cos(ωt)*i + b*sin(ωt)*j + 0*k, where i, j, and k are unit vectors in the x, y, and z directions, respectively. Again, we can plug in our known values for a, b, and ω, and solve for y and z at the given time, t.

Finally, it's important to note that the satellite will continue to move along its elliptical orbit, so the values of y and z will change over time. To find the specific values of y and z at any given time, we will need to use the equations above and plug in the appropriate time value.

## 1. How does Kepler's law explain the motion of a satellite in orbit?

Kepler's law states that a satellite in orbit around a planet will move in an elliptical path, with the planet at one focus of the ellipse. The satellite's speed will vary throughout its orbit, but the line connecting it to the planet will sweep out equal areas in equal times.

## 2. What factors influence the orbit of a satellite according to Kepler's laws?

The orbit of a satellite is influenced by the mass of the planet it is orbiting, the distance between the satellite and the planet, and the initial velocity of the satellite. These factors determine the shape, size, and speed of the satellite's orbit.

## 3. How does the distance from the planet affect the orbital period of a satellite?

According to Kepler's third law, the further away a satellite is from the planet it is orbiting, the longer its orbital period will be. This means that a satellite with a larger orbit will take longer to complete one full revolution around the planet compared to a satellite with a smaller orbit.

## 4. Can Kepler's laws be applied to all satellites in orbit, including man-made satellites?

Yes, Kepler's laws apply to all satellites in orbit, whether they are natural satellites like moons or man-made satellites launched by humans. They accurately describe the motion of objects in orbit around a central body.

## 5. How do scientists use Kepler's laws to study and predict the motion of satellites?

Scientists use Kepler's laws to calculate the orbital period, velocity, and distance of satellites in orbit. This information is crucial for predicting and planning satellite missions, as well as monitoring the health and performance of existing satellites in orbit.

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