(Solved): 1 Problem Johannes Kepler first discovered that the planets move in elliptical orbits around the Su ...

1 Problem Johannes Kepler first discovered that the planets move in elliptical orbits around the Sun. Satellites orbit the earth in a similar way. Consider the image below wherein a satellite is moving along an elliptical orbit with a semi-minor axis $a=13,860\mathrm{km}$ and semi-major axis $b=16,000$ $\mathrm{km}$ with the earth located at one focus of the ellipse with $c=8,000\mathrm{km}$. At the instant shown the satellite has a velocity $v=17,970\mathrm{km}/\mathrm{h}$. The acceleration of the satellite is purely due to gravity in the direction pointing from $A$ to the center of the earth. In general, gravitational acceleration follows an inverse square law. While the acceleration of gravity on the surface of the earth (radius of earth $r_e=6,371)$ has magnitude $g_e=9.821\mathrm{m}/{\mathrm{s}}^2$ the acceleration at point $A$ is $g_A=g_e{\left(r_e/r\right)}^2$. Derive a symbolic expression for the radius of curvature $?$ and the rate of change of speed $\stackrel{?}{v}$ of the orbit at $A$ as a function of the parameters of the problem using a path frame. (You can check your answer by plugging in the numerical values to confirm that $?=18,480\mathrm{km}$. Note: ${\hat{e}}_n$ is pointing straight down in this problem - not towards the center of the earth.) 2 Problem Repeat the problem above and determine $\stackrel{?}{r},\ddot{r},\stackrel{?}{?},\ddot{?}$ at the instant shown. Assume that $?$ is measured counter clockwise from the horizontal line passing through the center of the earth.