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.
|Use SI units consistently.|
|For ax and ay 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, vx, and vy in the first row. Let r = SQRT(x2+y2).|
|Use 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
to solve a differential equation numerically.)
|Use vx = vxo + axΔt
and vy = vy0 + ayΔt
with vxo , vyo, ax, and ay from
the previous row to find vx and vy in later
|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
|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.