Write the equations of motion for a $2$D projectile with drag. Given a coordinate

system where hat$(e{)}_1$ is horizontal and positive to the right and hat$(e{)}_2$ is vertical and

positive up$,$ let $x$ and $y$ be the horizontal and vertical degrees of freedom,

respectively. Model the drag force as

$F_{\text{d}\text{r}\text{a}\text{g}}=-\frac{1}{2}{C}_{D}A{v}^2(coshat(e{)}_1+sinhat(e{)}_2)$

recognizing that vcos$=x^{}$ and vsin$=y^{},$ the $x$ and $y$ components of drag

can be written as

$)x)y$

Incorporating the two equations above and the force due to gravity, you should

end up with two second$-$order equations of motion$-$one involving the horizontal

acceleration, $x^{},$ and one involving the vertical acceleration, $y^{}.$ Once you have

the two second$-$order equations, convert them to four first$-$order equations using

the method described in class. $($In class, we defined new variables, $y_1$ and $y_2.$

Here, call your new variables $u_1,u_2,u_3,$ and $u_4$ to avoid any confusion with the

degree of freedom $y.)$