Extra Credit 8

Consider an object moving in the gravitational field of the Earth.  Its acceleration is a = -GMEr/r3, where r is the position vector directed from the center of the Earth to the object.  Choose the origin of your coordinate system at the center of the Earth and assume the object is moving in the x-y plane.  Then the Cartesian components of the object's acceleration are

ax = -GMEx/(x2+y2)3/2,         ay = -GMEy/(x2+y2)3/2.

Write a spreadsheet or computer program to find the position of the object as a function of time.  Assume that the initial position of the object is at x = 0, y = 2RE, where RE is the radius of the Earth, and give the object an initial velocity of 5 km/s in the x direction.  The time increment should be made as small as practical.  Try 5 s.

Your spreadsheet should look similar to the one shown below.  You can use the attached spreadsheet as a starting point.

  A B C D E F G H
1 time x y ax ay vx vy r
2 0 0 1.27E+07 0 -2.45842 5.60E+03 0 12740000
3 5 2.80E+04 1.27E+07 -0.005403 -2.45841 5599.99 -12.29 12740000
4 10 5.60E+04 1.27E+07 -0.010806 -2.45839 5599.95 -24.58 12740000

To earn extra credit add your name and e-mail address to your spreadsheet and submit your spreadsheet for an initial speed of 5 km/s and for the initial speed corresponding to your most circular orbit.  In a few sentence describe how the kinetic, potential and total energy vary with time in those two cases.  What is the initial velocity for the most circular orbit?

Save your Excel spreadsheet (your name_ex8.xlsx), go to Canvas, Assignments, Extra Credit 8, and submit your spreadsheet.