homework2 - Astronomy 362 Problem Set#1 Due Friday January...

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Astronomy 362 Problem Set #1 Due Friday, January 24 at 12 noon In the homework problems below, “C&O” refers to your textbook. I have selected the problems with the expectation that they will take up to 1 hour of hard thinking/calculating each. If you are taking longer than this, feel free to come by my office to hash out where you are stuck. Note on collaborating: You may work together on this problem set, but all work presented here must be your own. You must clearly acknowledge any people you collaborated with. ASSUMED READING: Please finish C&O Chapters 2 and 3 before Friday, January 24. 1. [C&O 2.4 (tweaked)] Derive E = 1 2 μ v 2 GM μ r (Eq. 2.25), the expression for the total energy of a two-body system in terms of reduced mass from the expression for total energy of a two-body system in general cartesian coordinates E = 1 2 m 1 ! v 1 2 + 1 2 m 2 ! v 2 2 Gm 1 m 2 ! r 2 ! r 1 . HINT: Write down expressions for ! r 1 and ! r 2 in terms of ! r and the masses m 1 and m 2 first. Use those to find expressions for velocities ! v 1 = d ! r 1 dt and ! v 2 = d ! r 2 dt and go from there. You may assume constant masses. 2. [C&O 2.7 (tweaked)] Calculate the escape speed from the Solar System, starting from Earth's orbit. This is the minimum speed spacecraft must reach to escape directly from Earth to interstellar space (e.g. - New Horizons ). You can assume that the Sun constitutes all of the mass of the Solar System. NOTE: For full credit, clearly explain your reasoning.
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