## Relative Motion

### Notes for instructors

This module contains a sequence of sections, each of which deals with the motion of several objects. The motion is described first from the "standard" point of view -- i.e., of a "stationary" external observer. But often it is the case that motion is easier to understand, or simply must be observed, from the point of view of one of the moving objects. Situations of each type are presented here, mostly from planetary astronomy, in which the process of shifting one's reference frame becomes more and more complex.

Maple worksheets appear throughout the module, some simple animations appear on the main pages, and there are links from the second ("OJ") and fifth ("Mercury") parts to more involved animations.

To begin, there is a brief discussion of the use of parametric equations (x and y as functions of t) to describe the motion of particles. A first simple example, involving the motions of two objects thrown into the air from the ground, shows how to shift the point of view from the outside observer to that of one object or the other. This material requres no background in calculus at all, just familiarity with the equation of motion for an object under the influence of gravity.

An interesting problem involving relative velocity and (constant) acceleration follows. It involves a car passing a truck. Note that the car is a white Ford Bronco in the animation of the situation (which gives a hint as to the vintage of the problem). This part could use some simple calculus (velocity and acceleration as derivatives) or simply the equations for motion under constant acceleration as they are derived in many (non-calculus-based) physics courses.

The next part moves into two dimensions. It concerns the orbit of Mars and of other planets, first viewed from the heliocentric point of view, then from the point of view of an observer on one of the planets (Earth, for example). No calculus is required for this -- in fact it gives an interesting natural occurrence of polar coordinate curves (looped cardioids, etc). After doing these problems, it is interesting to discuss the issue of why it took so long to move the center of our universe from the Earth to the Sun.

The fourth part adds rotation to the types of motion considered. It concerns the motions of the Moon and the Earth as viewed from each other. It can be studied either after students have encountered rotation matrices in linear algebra, or in calculus when rotation of axes is discussed. Once students ascertain that the earth is motionless in the sky above the moon (but lunar residents would observe the rotation of the earth), it becomes interesting to discuss how astronomy might have evolved as seen from the moon.

The final part of the module, which has no prerequisites beyond the preceding part, concerns the peculiar apparent motion of the sun as seen from Mercury (due to the eccentricity of Mercury's orbit and the fact that its days are very long compared to its years). The animation at the end provides an interesting view of this phenomenon.