A question about a moving satellite (Kepler)

[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?

Homework Equations

The Attempt at a Solution

 

[tms]

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.

 

[sapz]

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

 

[haruspex]

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

 

[Nickie]

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.