306 HW 2 - Consider an Earth orbit of a =7000km and e =0.2...

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1 MAE 306 Fall 2008 HW 2 Due 5:00 PM, Feb. 7, 2008 Vector Kinematics Consider three orthonormal basis vectors k j i ˆ , ˆ , ˆ that comprise coordinate system N. Let k j i a ˆ 5 ˆ 4 ˆ 3 + + = . 1. (2 points) What is the projection of a onto j ˆ ? 2. (2 points) Find N a . 3. (2 points) Calculate | a |. 4. (2 points) Show that 0 a a = × Consider three other orthonormal basis vectors t s r ˆ , ˆ , ˆ that comprise coordinate system M. N is rotated relative to M by a rotation about the axis i ˆ by an angle 6 / π , as shown in the figure. Let t s r b ˆ 5 ˆ 4 ˆ 3 + + = 5. (2 points) Write down j ˆ in terms of t s r ˆ , ˆ , ˆ . 6. (2 points) Find M a . 7. (2 points) Calculate b a × and represent the answer in M axes. 8. (2 points) Let t a b c ˆ 5 × = . Find N c . 9. (2 points) Calculate b a . Orbit Mechanics Consider a circular solar orbit at a distance of 1,000,000 km. 10. (2 points) Calculate the orbit’s eccentricity. 11. (2 points) Calculate its specific energy. 12. (2 points) Calculate its period.
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Unformatted text preview: Consider an Earth orbit of a =7000km and e =0.2 13. (2 points) Calculate the mean motion. 14. (2 points) Calculate the radius at perigee. 15. (2 points) Calculate the velocity at apogee. 16. (2 points) Calculate the velocity at perigee of a spacecraft in a parabolic Jovian (i.e. Jupiter) orbit whose closest approach to the planet is 100,000 km. Proposal Questions 17. (2 points) Show the steps in Arthur C. Clarkes correct calculation of the worst-case eclipse duration for a geostationary spacecraft. 18. (2 points) Calculate a , e , E , n , and the period for two of the orbits in which your spacecraft will find itself during its lifetime. If you can think of only one orbit, also calculate the kinetic energy (not the specific energy) and the velocity at true anomaly ( nu ) =- /4 for that single orbit....
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