(a) Express in terms of rmin and vmax the angular momentum

Chapter 13, Problem 13.118

(choose chapter or problem)

(a) Express in terms of \(r_{\min }\) and \(v_{\max }\) the angular momentum per unit mass, h, and the total energy per unit mass, E/m, of a space vehicle moving under the gravitational attraction of a planet of mass M (Fig. 13.15). (b) Eliminating \(v_{\max }\) between the equations obtained, derive the formula

\(\frac{1}{r_{\min }}=\frac{G M}{h^{2}}\left[1+ _{\mathrm{B}} \overline{1+\frac{2 E}{m}\left(\frac{h}{G M}\right)^{2}}\right]\)

(c) Show that the eccentricity £ of the trajectory of the vehicle can be expressed as

\(\mathrm{e}=_{\mathrm{B}} \overline{1+\frac{2 E}{m}\left(\frac{h}{G M}\right)^{2}}\)

(d) Further show that the trajectory of the vehicle is a hyperbola, an ellipse, or a parabola, depending on whether E is positive, negative, or zero.