Problem $3(30$ points$)$

Your satellite has a sensor that needs to be pointed at a target spacecraft. So you need the position, velocity, and

acceleration of the target relative to your satellite. At a given instant, the ECI $($intertial$)$ position, velocity and

acceleration of your satellite $($the origin of a moving frame$)$ are

vec$(r{)}_0=7178\mathrm{.}050$widehat$(x)-32\mathrm{.}616$widehat$(Y)-14\mathrm{.}170$widehat$(Z)(km)$

vec$(v{)}_0=0\mathrm{.}034$widehat$(x)+5\mathrm{.}269$widehat$(Y)+5\mathrm{.}269$widehat$(Z)(k\frac{m}{s})$

vec$(a{)}_0=-7\mathrm{.}8{10}^{-3}$widehat$(x)+3\mathrm{.}5{10}^{-5}$widehat$(Y)+1\mathrm{.}5{10}^{-5}$widehat$(Z)(k\frac{m}{s^2})$

The angular velocity and acceleration of the moving frame relative to the inertial frame are

vec$()=0\mathrm{.}03$hat$(x)-0\mathrm{.}04$hat$(y)+0\mathrm{.}10$hat$(z)(ra\frac{d}{s})$

vec$({)}^{}=-0\mathrm{.}005$hat$(x)+0\mathrm{.}003$hat$(y)-0\mathrm{.}011$hat$(z)(ra\frac{d}{s^2})$

The inertial position, velocity, and acceleration of your target $(P)$ are

$r^{}=6757\mathrm{.}650$widehat$(x)+1281\mathrm{.}720$widehat$(Y)-13\mathrm{.}342$widehat$(Z)(km)$

$v^{}=1\mathrm{.}014$widehat$(x)-5\mathrm{.}289$widehat$(Y)+5\mathrm{.}381$widehat$(Z)(k\frac{m}{s})$

vec$(a)=8\mathrm{.}3{10}^{-3}$widehat$(x)+1\mathrm{.}6{10}^{-3}$widehat$(Y)+1\mathrm{.}6{10}^{-5}$widehat$(Z)(k\frac{m}{s^2})$

The unit vectors of your satellite $($moving$)$ frame at this instant in time are

widehat$(x)=1\mathrm{.}0000$widehat$(x)-0\mathrm{.}0045$widehat$(Y)-0\mathrm{.}0020$widehat$(Z)$

hat$(y)=0\mathrm{.}0046$widehat$(x)+0\mathrm{.}7071$widehat$(Y)+0\mathrm{.}7071$widehat$(Z)$

hat$(z)=-0\mathrm{.}0018$widehat$(x)-0\mathrm{.}7071$widehat$(Y)+0\mathrm{.}7071$widehat$(Z)$

Show that vec$(v{)}_{\text{r}\text{e}\text{l}}=56\mathrm{.}708$hat$(x)+7\mathrm{.}409$hat$(y)-3\mathrm{.}248$hat$(z)(k\frac{m}{s}).$

Find the relative acceleration vec$(a{)}_{\text{r}\text{e}\text{l}}$ of P using moving frame coordinates.