Chapter 13.2, Problem 13.112P

Show that the values v_A and v_P of the speed of an earth satellite at the apogee A and the perigee P of an elliptic orbit are defined by the relations

$\begin{array}{ll}{v}_A^2=\left(\frac{2GM}{r_A+r_P}\frac{r_P}{r_A}\right)\hfill & v_P^2=\frac{2GM}{r_A+r_P}\frac{r_A}{r_P}\hfill \end{array}$

where M is the mass of the earth, and r_A and r_P represent, respectively, the maximum and minimum distances of the orbit to the center of the earth.

Fig. P13.112 and P13.113