x(t)=R cos (
t),
y(t)=R sin (t),
R = 6,760,000 meters and
= 0.001139905 radians/second
(t is in seconds).
Copyright 1997 by Mark Janeba
Planet X's center is at (0,0) in the Universal Coordinate system. An orbiting space station has position
r(t)=(x(t),y(t)),
with
x(t)=R cos (t),
y(t)=R sin (t),
R = 6,760,000 meters and
= 0.001139905 radians/second
(t is in seconds).
Discuss the station's orbit, velocity vector, speed, and acceleration. Show that the station's acceleration is a vector pointing from the station to the center of planet X, and the acceleration is equal to
for some constant G.
Next, imagine that at t=0, an astronaut on the station throws a beer can pretty much directly at planet X. Since there is a near-vacuum in orbit, there is no air resistance to slow this bit of litter (at least not until it hits the atmosphere, about 161,000 meters below). A rough approximation of the can's position is rc(t) = (xc(t), yc(t)), with
Confirm that for t near 0, the can and astronaut are in close proximity (explain). How fast did the astronaut throw the can? In precisely what direction?
Assuming that the approximation for the can's position is valid for large values of t: What is the can's long-term behavior? What is the closest the can gets to the planet? Does the can reach the atmosphere? Suppose the astronaut watches (or tries to watch) the can with a telescope for 20 minutes, and then works outside for another 80 minutes. What is the closest the can ever gets to the astronaut while s/he is outside but not watching? Where is it at that time?
From the astronaut's point of view, describe the can's apparent motion. (Recall the astronaut feels that s/he is motionless, with the planet turning below him/her).
Finally, compare the can's acceleration at time t with 
for
several values of t. This tests the quality-of-fit for the approximated
position functions for the can; the can's true position function has acceleration
exactly equal to for
all values of t with the same G as you found at the beginning
of the project.
Due Wednesday, December 6th, at 5 p.m. at Smullin 220.
willamette.edu