Chapter 13.2, Problem 13.118P

(a)

To determine

Express energy per unit mass equation in terms of $\left(r_{\min }\right)$ and $\left(v_{\max }\right)$ of a space vehicle moving under the gravitational attraction of a planet of mass M.

(b)

To determine

Derive the formula $\frac{1}{r_{\min }}=\frac{GM}{h^2}\left[1+\sqrt{1+\frac{2E}{m}{\left(\frac{h}{GM}\right)}^2}\right]$ by eliminating $\left(v_{\max }\right)$.

(c)

To determine

Show the eccentricity $\left(\epsilon \right)$ of the trajectory of the vehicle can be expressed as, $\epsilon =\sqrt{1+\frac{2E}{m}{\left(\frac{h}{GM}\right)}^2}$ by eliminating $\left(v_{\max }\right)$.

(d)

To determine

Show the trajectory of the vehicle is hyperbola, ellipse or parabola depending on whether E is positive, negative or zero respectively.