In the following problems, except where otherwise indicated, assume that the radius of the earth is $6378 \mathrm{km}$, the earth's mass is $5.976\left(10^{24}\right) \mathrm{kg}$, the mass of the sun is $1.99\left(10^{30}\right) \mathrm{kg}, \quad$ and $\quad$ the $\quad$ gravitational $\quad$ constant $\quad$ is $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$.
A satellite is launched at its apogee with an initial velocity $v_{0}=2500 \mathrm{mi} / \mathrm{h}$ parallel to the surface of the earth. Determine the required altitude (or range of altitudes) above the earth's surface for launching if the free-flight trajectory is to be (a) circular,
(b) parabolic,
(c) elliptical, with launch at apogee, and
(d) hyperbolic. Take
\[
G=34.4\left(10^{-9}\right)\left(\mathrm{lb} \cdot \mathrm{ft}^{2}\right) / \mathrm{slug}^{2}, \quad M_{e}=409\left(10^{21}\right) \mathrm{slug}, \quad \text { the }
\]
earth's radius $r_{c}=3960 \mathrm{mi},$ and $1 \mathrm{mi}=5280 \mathrm{ft}$.