Write a Matlab script that starts with position and velocity vectors $($for an Earth orbit$)$ in matrix form $($ECI frame$)r_1^E$ and $v_1^E$ at time $t_1,$ and determines $r_2^E$ and $v_2^E$ at time $t_2($in the ECI frame$).$ Using methods developed in class and applied in homework and code blocks, you will need to calculate the values of the orbital elements, solve a Kepler problem, determine vec$(r{)}_2$ and vec$(v{)}_2$ in perifocal coordinates, and then transform those to the ECI frame.

Use the values

$r_1^E=\left[\begin{array}{c}-4357\mathrm{.}1\\ 7885\mathrm{.}8\\ -2225\mathrm{.}3\end{array}\right],(km),v_1^E=\left[\begin{array}{c}-4\mathrm{.}224\\ -5\mathrm{.}718\\ -4\mathrm{.}864\end{array}\right],(k\frac{m}{s}),t_1=1564s,t_2=32548s,{}_{o+}=3\mathrm{.}986{10}^{5}k\frac{m^3}{s^2}$

Required:

$($Proj $2$a and Proj $2$b$)$ You must work in groups of $2.$ You must use the Project Groups in Canvas $-$ this assignment is linked to those groups and will not record grades for other groups that you create.

Proj $2$a $($Do this part first, before you write the code.$)$ A report $($typed or handwritten$)$ showing the mathematical analysis used. This is the solution in terms of relevant equations and procedures. It is a list of equations, algorithms, procedures, etc. in the order that your code uses them. It is NOT a narrative description such as "First, we took the cross$-$product of vec$(r{)}_1$ and vec$(v{)}_1$ to get angular momentum vec$(h),$ then we calculated the energy $..."$ For an example, see the posted solution in Canvas for Proj $1$b$.$

$($a$)$ Do not include numerical calculations in this report.

$($b$)$ Include both group members' names, your group number and the integrity statement.

Proj $2$b Matlab code. Your code must:

$3.$ be a single file $($either $.$m or $.$mlx $)$ that contains the script, with any functions appended to the end of the script. Do NOT put your functions in separate files.

$4.$ contain a comment line $($at the top$)$ with the names of both members of your group and your group number

$5.$ contain a comment line $($at the top$)$ with the integrity statement

$6.$ contain a line $($at the top$)$ with the commonds: clc; clear all; close all;

$7.$ for any functions, use the form function $[$outputs$]=$ myfunction$($ inputs $)$

$8.$ not use keyboard or file inputs. The values of $r_1^E,v_1^E,t_1$ and $t_2$ must be "hardwired" into the code $($i$.$e$.,$ their values must be set with assignment statements such as alpha $=124$;$.$

$9.$ calculate and output the values of $a,e,i,,,t_p.$

$10.$ calculate and output the DCM $C^{EP}$

$11.$ calculate and output the position and velocity vec$(r{)}_2$ and vec$(v{)}_2$ in ECI coordinates at time $t_2$

$($continued on next page$)$

$12.$ include labels and units $($as appropriate$)$ for all output quantities

$13.$ Use the value $={10}^{-4}$ for the Newton$-$Raphson iteration.

Note regarding the output: it's ok to print the vectors in matrix form as row matrices, e$.$g$.,r_{-1}=[500\mathrm{.}2,600\mathrm{.}3,742\mathrm{.}0](km).$ Don't forget the label $($e$.$g$.,$ r$\_1$E$)$ and the units $($e$.$g$.,($km$)).$