Let G denote the universal gravitational constant and let Mand m denote masses a

Chapter 12, Problem 1

(choose chapter or problem)

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

radius of Earth = 4000 mi = 6440 km

radius of Moon = 1080 mi = 1740 km

1 year (Earth year) = 365 days

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

(b) Use Formulas (7), (17), and (22) to show that

       \(v \times b=G M[(e+\cos \theta) i+\sin \theta j]\)

(c) Show that \(\|v \times b\|=\|v\|\|b\|\)

(d) Use the results in parts (b) and (c) to show that the speed of a particle in an elliptical orbit is

          \(v=\frac{v_{0}}{1+p} \sqrt{e^{2}+2 e \cos \theta+1}\)

(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.

Equation Transcription:

Text Transcription:

t=0  

v × b = GM[(e + cos θ )i + sin θ j]  

||v × b||=||v|| ||b||

v=v_0/1+e sqrt e^2 +2e cos θ + 1