A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the center of the Earth, $G$ is the universal gravitational constant. Consider the initial data $R(0)=(0,0,R_e+h),\dot R(0)=v$, where $R_e$ is the radius of the Earth.
(a) Regarding $\epsilon=h/R_e$ as a small parameter use asymptotic expansions to derive the ever-boring equation of ballistic motion $\ddot r=-g, r(0)=(0,0,h),\dot r(0)=v.$
(b)Derive the first order correction for the ballistic motion.
(a)I tried to use $r=R-R_e(0,0,1)$ and regard $h$ and $|v|$ to be $O(1)$ quantities. We also know $g=\frac{GM_e}{R^2}$. Then (a) seems obvious to me. I don't know where I should use asymptotic expansions.
(b)I really don't know how to do the first order correction.
Could someone kindly offer some help? Thanks!