# Very difficult kinematics problem involving spherical coordinates

1. Nov 11, 2008

### x2carlos

1. The problem statement, all variables and given/known data

A satellite is in motion over the Earth. The Earth is modeled as a sphere of radius R that rotates with constant angular velocity "$$\Omega$$" in the direction of Ez, where Ez, lies in the direction from the center of the Earth to the North Pole of the Earth at point N. The position of the satellite is known geographically in terms of its radical distance, r, from the center of the Earth, its EARTH RELATIVE longitude, "$$\theta$$", where "$$\theta$$" is the angle measured from direction Ex, where Ex lies along the line from the center of the Earth to the intersection of the Equator with the Prime Meridian, and its latitude, "$$\phi$$", where "$$\phi$$" is measured from the line that lies along the projection of the position into the equitorial plane.

Using spherical basis (Er,E"theta", E"phi") to the describe the position of the spacecraft (where Er=direction r from center of earth to spacecraft, E"$$\theta$$"=direction of Ez x Er, and E"$$\phi$$"= Er x E"$$\theta$$"), determine the velocity and acceleration of the satellite a) as viewed by and observer fixed to the earth b) as viewed by an observer fixed to an inertial refference frame.

2. Relevant equations

transport theorem

3. The attempt at a solution

I established 3 reference frames: one inertial fixed to Ex, Ey, Ez where Ey is Ez x Ex all at t=0
second one the same but fixed to earth, so it rotates with angular velocity "$$\Omega$$"
third , in the direction of r, (spherical coordinate system)

I am not sure if those are the correct ones but with those I am getting and angular velocity
("$$\theta$$dot" + $$\Omega$$)Ez - "$$\phi$$dot" E"theta"

Well I understand that it might be hard to visualize what is going on but I dont know how to upload the figure that corresponds. My main problem with spherical coordinates is that they are hard for me to visualize and in this particular problem im having trouble determining what the angular velocity of the space craft is relative to an inertial reference frame in order to apply the transport theorem.

I dont know how to attach my full solution but that my main problem and i think the rest of my crazy algebra problem stem from that, I am basically wondering if there is an easier way to set the problem up to ease the algebra involved
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution