Please show all the work Problem 3: 20 points] The location of a vehicle is desc

Question

Please show all the work

Problem 3: 20 points] The location of a vehicle is described by its longitude from Greenwich, its latitude 8, and its altitude H above the surface of the earth as shown in Figure 3. Of course, the earth itself rotates at angular velocity about its polar axis. A coordinate frame s has its origin at the center of the earth, with the s vector in the plane of the equator, the s vector pointing at the vehicle and s, completing the right handed set. Obtain an expression for the inertial velocity of the vehicle. Re s2 Figure 3: Flight vehicle located by altitude, longitude, and latitude with respect to the rotating earth

Explanation / Answer

given lambda = longitude
del = latitude
altitude = H
angular speed of earth = we

hence in the fram eof reference given
the locatino of the vehicle is given by

in the frame of reference where earth is at rest ( ibnertial frame)

location
r = (Re + H)cos(del)cos(lambda)i + (Re + H)cos(del)sin(lambda)j + (Re + H)sin(del)k
then
v = r' = [[H']cos(del)cos(lambda) - [Re + H]sin(del)cos(lambda)*del' - [Re + H]cos(del)sin(lambda)lambda' ]i + [[H']cos(del)sin(lambda) - [Re + H]sin(del)sin(lambda)del' + [Re + H]sin(del)sin(lambda)lambda']j + [[H']sin(del) + [Re + H]*cos(del)del']k

here H' is rate of chance of altitude of satellite in s2 direction
del' = rate of change of latitude of satellite
lambda' = rate of change of longitude
if non inertial angular speed of craft in this direction is w2, then lambda' = w2 - we

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