Spreadsheet Exercise

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. Than 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 5km/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.

  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

 

bulletUse SI units consistently. 
bulletFor ax and ay use the formulas given above, and let them refer to the x and y values in the same row.
bulletUse the given initial conditions for x, y, vx, and vy in the first row.  Let r = SQRT(x2+y2).
bulletUse x = x0 + vxΔt and y = y0 + vyΔt with x0 , y0, vx, and vy from the previous row to find x and y in later rows.  (You are using the  Euler method to solve a differential equation numerically.)
bulletFor example, in the spreadsheet shown above, type =B2+F2*(A3-A2) into cell B3.
bulletFor smaller numerical errors, use x = x0 + vxΔt + (1/2)axΔt2 and y = y0 + vyΔt + (1/2)ayΔt2.  For example, type =B2+F2*(A3-A2)+0.5*D2*(A3-A2)^2 into cell B3.
bulletUse vx = vxo + axΔt and vy = vy0 + ayΔt with vxo , vyo, ax, and ay from the previous row to find vx and  vy in later rows.
bulletFor example, in the spreadsheet shown above, type =F2+D2*(A3-A2) into cell F3.
bulletFor smaller numerical errors, use vx = vxo+axΔt+(1/2)(Δax/Δt)Δt2 and vy = vy0 + ayΔt + (1/2)(Δay/Δt)Δt2.   For example, type =F2+D2*(A3-A2)+0.5*(D3-D2)*(A3-A2) into cell F3.
bulletIf you use 5 s time intervals you need to copy your formulas down approximately 3000 rows for one complete revolution.
bulletProduce a scatter plot of y versus x similar to the plot shown below.

bulletVary the magnitude of the initial velocity until a circular orbit is found.
bulletIf r becomes smaller than the radius of the earth, the objects hits the earth.
bulletAdd 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 this most circular orbit?

Save your Excel document (your name_exm8.xls).  Go to Blackboard, Assignments, Extra Credit 8, and attach your document.