Find the equations of motion for the asteroid in Problem, as seen from Earth frame of reference described in that problem, and solve them numerically using Mathematica. As starting conditions, assume that die asteroid is in direct opposition to the Sun relative to Earth at t = 0, and use initial velocities appropriate to circular orbits about die Sun to calculate die initial velocity of die asteroid relative to Earth. Plot the trajectory of the satellite for one orbital period.
A new asteroid is discovered in a circular orbit or radius 41/3 AU about the Sun,4 Its period of revolution is precisely 2 years. Assume that at t = 0 it is at its distance of closest approach to the Earth.
(a) Find its coordinates [x(t), y(t)] in a frame of reference fixed to Earth but whose axes remain fixed in orientation relative to the Sun. Assume that the x-axis points toward tire asteroid at t = 0.
(b) Calculate tire velocity of the asteroid, relative to Earth, at t = 0.
(c) Find the x- and y-components of the acceleration of the asteroid in dais frame of reference. Integrate them twice to show that tire resulting positions as a function of time agree with part (a).
(d) Plot the trajectory of the asteroid as seen from die rotating frame of reference attached to Earth for its 2-year orbital period. (Hint: Use Mathematica’s ParametricPlot graphing tool.)
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