Consider an object moving in the gravitational field of the Earth. Its acceleration is

a_{x }= -GM_{E}x/(x^{2}+y^{2})^{3/2}_{,
}a_{y }= -GM_{E}y/(x^{2}+y^{2})^{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 | 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(x^{2}+y^{2}). | |||||

Use x = x_{0 }+ v_{x}Δt
and y = y_{0 }+ v_{y}Δt
with x_{0} , y_{0}, 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.)
| |||||

Use v_{x }= v_{xo }+ a_{x}Δt
and v_{y }= v_{y0 }+ a_{y}Δt
with v_{xo} , v_{yo}, a_{x}, and a_{y} from
the previous row to find v_{x} and v_{y} in later
rows.
| |||||

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.
| |||||

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 this most circular orbit?

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