Consider an object moving in the gravitational field of the Earth.  Its acceleration is a = -GM_Er/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

a_x = -GM_Ex/(x²+y²)^3/2_,         a_y = -GM_Ey/(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 = 2R_E, where R_E 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.

• Use SI units consistently. 
• For a_x and a_y use the formulas given above, and let them refer to the x and y values in the same row.
• Use the given initial conditions for x, y, v_x, and v_y in the first row.  Let r = SQRT(x²+y²).
• Use x = x₀ + v_xΔt and y = y₀ + v_yΔt with x₀ , y₀, v_x, and v_y from the previous row to find x and y in later rows.  (You are using the  Euler method to solve a differential equation numerically.)
  • For example, in the spreadsheet shown above, type =B2+F2*(A3-A2) into cell B3.
  • For smaller numerical errors, use x = x₀ + v_xΔt + ½a_xΔt² and y = y₀ + v_yΔt + ½a_yΔt².  For example, type =B2+F2*(A3-A2)+0.5*D2*(A3-A2)^2 into cell B3.
• Use v_x = v_xo + a_xΔt and v_y = v_y0 + a_yΔt with v_xo , v_yo, a_x, and a_y from the previous row to find v_x and  v_y in later rows.
  • For example, in the spreadsheet shown above, type =F2+D2*(A3-A2) into cell F3.
  • For smaller numerical errors, use v_x = v_xo+a_xΔt+½(Δa_x/Δt)Δt² and v_y = v_y0 + a_yΔt + ½(Δa_y/Δt)Δt².   For example, type =F2+D2*(A3-A2)+0.5*(D3-D2)*(A3-A2) into cell F3.
• If you use 5 s time intervals you need to copy your formulas down approximately 3000 rows for one complete revolution.
• Produce a scatter plot of y versus x similar to the plot shown below.

• Vary the magnitude of the initial velocity until a circular orbit is found.
  • If r becomes smaller than the radius of the earth, the objects hits the earth.
• Add three columns to your spreadsheet to calculate the kinetic energy, potential energy and total energy as a function of time.  (Set the mass of orbiting the object equal to 1 kg.)  How do they vary with time?  (Note the potential energy and the total energy will be negative.)

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.