Chapter 12.7, Problem 1ES

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

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

(a) Obtain the value of C given in Formula (16) by setting $t=0\text{in}\text{(15)}\text{.}$

(b) Use Formulas (7), (17), and (22) to show that $\text{v}\times \text{b}=GM[(e+\cos \theta )\text{i}+\sin \theta \text{j]}$

(c) Show that $‖\text{v}\times \text{b}‖=‖\text{v}‖‖\text{b}‖.$

(d) Use the result in parts (b) and (c) to show that the speed of a particle in an elliptical orbit is $\upsilon =\frac{{\upsilon }_0}{1+e}\sqrt{e^2+2e\cos \theta +1}$

(e) Suppose that a particle is in an elliptical orbit. Use part (d) to conclude that the distance from the particle to the center of force is a minimum if and only if the speed of the particle is a maximum. Similarly, argue that the distance from the particle to the center of force is a maximum if and only if the speed of the particle is a minimum.