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To consider motion in more than one dimension, we specify both the *x* and *y* coordinates of an object as functions of time. For example, for an object that is undergoing circular motion -- say its circular path has radius 5 and is centered at the origin, and it traverses a complete circle every 20 seconds -- we can express its motion as

x(t) = 5 cos (pt / 10); y(t) = 5 sin (pt / 10);

Here is an animation of the motion, produced by the Maple statement:

animate([5*cos(Pi*t/10)+0.5*cos(s),5*sin(Pi*t/10)+0.5*sin(s),s=0..2*Pi],t=0..20);

To calculate relative motion in two dimensions, whether relative position, velocity, or acceleration, one proceeds as in one dimension, by subtracting the positions, velocities, or accelerations of the objects in question. Of course, you have to be careful to subtract *x*-coordinates from *x*-coordinates and *y*-coordinates from *y*-coordinates.

The most famous relative motion problem of all concerns the motion of the planets. Astronomers and astrologers realized that the planets moved against the backdrop of "fixed stars," but according to a pattern that seemed very mysterious -- most of the time planets move in one direction among the stars, but occasionally halt their motion and even regress a little from time to time. Ancient philosophers from Ptolemy to Aristarchus attempted to explain the peculiar motions of the planets -- mostly without complete success. Let's explore why this problem is so hard.

Consider the motion of Mars in the sky. We know now that the Earth circles the sun once a year in a roughly circular orbit that is about 93 million miles in radius. Mars is about 1.5 times as far from the sun as the earth, and takes about 2 Earth years to go once around the sun. For the moment, we assume that the orbits of the Earth and Mars are circular, with the sun at the center, and that the figures just given for the orbit of Mars are precise.

Then the Earth's orbit -- with distance measured in "astronomical units" (1 a.u. = 93,000,000 miles) and with time measured in years, is given by the equations:

xE := cos(2*Pi*t); yE := sin(2*Pi*t);

These are the equations of a circle of radius 1 (note that *x*_{E}^{2} + *y*_{E}^{2 } = 1) that is traversed once every 1 time unit.

Mars's orbit -- of radius 1.5 and period 2 -- is given by:

xM := 1.5*cos(Pi*t); yM : = 1.5*sin(Pi*t);

As before, we can produce animations of these orbits together. Note that things are back where they started every 2 time units (2 years).

animate({[xE+0.1*cos(s),yE+0.1*sin(s),s= 0..2*Pi],

[xM+0.1*cos(s),yM+0.1*sin(s),s=0..2*Pi]},t=0..2);

For reference, here is the Maple worksheet.

This animation gives a picture relative to the sun -- the sun is at the center and remains there for the entire time. But until very recently, humans have been constrained to observe the heavens from the earth. So let's look at the motion of Mars relative to the earth:

xR := xM - xE; yR := yM - yE;

Look here for the solution if you need help.

One good Web source for planetary data is NASA, in particular their Welcome to the planets site at CalTech's Jet Propulsion Lab.

**Problem 7:** Jupiter is about five times as far from the sun as the earth, and it takes about 12 earth years for Jupiter to go once around the sun. (You can use the NASA site for more precise information). Repeat the whole analysis for Jupiter. How often does Jupiter exhibit retrograde motion?

Larry Gladney and Dennis DeTurck, "Relative Motion - Relative motion in two dimensions," *Loci* (November 2004)

Journal of Online Mathematics and its Applications