Spreadsheet Exercise

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. Than 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 = 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 5s.

Your spreadsheet should look similar to the one shown below.

	A	B	C	D	E	F	G	H
1	time	x	y	a_x	a_y	v_x	v_y	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

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_xDt and y = y₀+ v_yDt with x₀ , y₀, v_x, and v_y from the previous row to find x and y in later rows. See Serway, page 160 for a discussion of the Euler method.

	For example, in the spreadsheet shown above, type =B2+F2*(A3-A2) into cell B3.
	For smaller numerical errors, use x = x₀+ v_xDt + (1/2)a_xDt² and y = y₀+ v_yDt + (1/2)a_yDt². For example, type =B2+F2(A3-A2)+0.5D2*(A3-A2)^2 into cell B3.

Use v_x= v_xo+ a_xDt and v_y= v_y0+ a_yDt 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_xDt+(1/2)(Da_x/Dt)Dt² and v_y= v_y0+ a_yDt + (1/2)(Da_y/Dt)Dt². For example, type =F2+D2(A3-A2)+0.5(D3-D2)*(A3-A2) into cell F3.

If you use 5s 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 1kg.) 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 5km/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) and attach it to an e-mail messageto your instructor.

If you prefer to use another software package or programming language instead of Excel, you are welcome to do so. (You are also welcome to use more advanced numerical methods if you are familiar with them.) You must submit your program, plots, and answers to earn extra credit points.