In all the exercises below, show your work and simplify the final answer as much
ID: 2123667 • Letter: I
Question
In all the exercises below, show your work and simplify the final answer as much as possible. Never leave the answer in terms of sin or cos /2
(A) What kind of curve is the orbit?
(B) State Kepler's three laws of planetary orbits.
(C) Determine x(%u03C8) and y(%u03C8).
(D) Determine %u03BE(%u03C8) and %u03B7(%u03C8).
(E) Express r as a function of .
(F) The angular momentum is L = m r2 d%u03C6/dt. Determine L in terms of {a,e,T} Hence verify that L is a constant of the motion.
(G) The energy is E=[1/2]m(dr/dt)2+[1/2]mr2 (d%u03C6/dt)2%u2212GMm/r.
Determine E in terms of {a,e,T}.
(H) E must be a constant of the motion. Hence determine T and E from the equation that you obtained in (G).
(I) Write T in terms of the spatial orbit parameters and the force constant GM.
(J) Now express L and E in terms of a, e, GM.
(K) In Figure 1, determine the Cartesian coordinates of the points P, A, and R
(L) In Figure 1, determine the time t at the point R.
Figure 1 illustrates a Keplerian orbit, with Cartesian coordinates (x, y) and polar coordinates (r, ).Explanation / Answer
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