Chapter 12.7, Problem 5ES

In exercises that require numerical values, use Table 12.7.1 and the following values, where needed:

$\begin{array}{l}\text{radius}\text{of}\text{Earth}=4000\text{mi}=6440\text{km}\\ \text{radius}\text{of}\text{Moon}=1080\text{mi}=1740\text{km}\\ \text{1year (Earth Year)}=365\text{days}\end{array}$

Suppose that a particle is in an elliptical orbit in a central force field in which the center of force is at a focus, and let $\text{r}=\text{r(}t\text{) and v}=\text{v}(t)$ be the position and velocity functions of the particle, respectively. Let $r_{\min }\text{and}{r}_{\max }$ denote the minimum and maximum distances form the particle to the center of force, and let $v_{\min }\text{and}{\upsilon }_{\max }$ denote the minimum and maximum speeds of the speeds of the particle.

(a) Review the discussion of ellipses in polar coordinates in Section 10.6, and show that if the ellipse has eccentricity e and semimajor axis a, then $r_{\min }=a(1-e)\text{and}{r}_{\max }=a(1+e).$

(b) Explain why $r_{\min }\text{and}{\text{r}}_{\max }$ occur at points at which r and v are orthogonal.

(c) Explain why ${\upsilon }_{\min }\text{and}{\upsilon }_{\max }$ occur at points at which r and v are orthogonal.

(d) Use Equation (2) and parts (b) and (c) to conclude that $r_{\max }{\upsilon }_{\min }=r_{\min }{\upsilon }_{\max }.$