----- The following copyright 1991 by Dirk Terrell
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 Lecture #4 "

   A contemporary of Galileo was Johannes Kepler and the two of them were 
instrumental in the search for the understanding of planetary motion. Kepler 
was a man of considerable mathematical abilities, and using the 
observational data of Tycho Brahe, he formulated three laws to describe 
planetary motion. We now refer to these laws as Kepler's Laws and they can 
be stated as follows:

  First Law:  The orbits of the planets are in the shapes of ellipses with the
              sun at one focus. (Take some time to review the geometry of the
              ellipse if you don't know what the focus is. Remember the 
              "string method" for drawing an ellipse? Put two tacks into a 
              piece of paper, tie a string into a loop, put the string around
              the tacks and pull it tight with a pencil. Now draw a "circle"
              keeping the string tight and you will have an ellipse. The places
              where the tacks are placed are called the foci.)
  Second Law: A line connecting the planet to the sun sweeps out equal areas
              in equal time periods, no matter where the planet is in its
              orbit. Another way to state it is that a planet moves faster
              when it is closer to the sun and slower when it is far away 
              from the sun.
  Third Law:  The period of a planet's orbit is related to its distance
              from the sun. We measure the size of an elliptical orbit by its
              semi-major axis. (The semi-major axis of an ellipse is half the 
              length of the line that cuts the ellipse in half in its longest
              dimension. For example, take a loaf of french bread, which has
              a somewhat elliptical cross section, and cut it in half from one
              end to the other, and you would be cutting along the semi-major
              axis. Again, get the encyclopedia out and review the geometry of
              the ellipse if this is greek to you.) If the period is measured
              in years and the semi-major axis in astronomical units (A.U., the
              distance from the earth to the sun), the relationship is rather
              simple - the period squared equals the semi-major axis cubed. As
              and equation we can write it as P^2=a^3 or P*P=a*a*a.

   Kepler's Laws are empirical laws, because he derived them from data that 
had been taken. He had no idea WHY period squared equalled semi-major axis 
cubed. He just knew that it fit the data. The explanation of why Kepler's 
Laws worked came a few years later in the work of perhaps the most 
influential scientist ever- Sir Isaac Newton.

   Newton's great work was the book "Philosophiae Naturalis Principia 
Mathematica", sometimes just referred to as the Principia. The central key 
to the Principia was the idea of universal gravitation- every object in the 
universe attracts every other object. The force of attraction is 
proportional to the masses of the objects and inversely proportional to the 
square of the distance between them. Think about that inverse square idea. 
It means that if you measure the attraction between two objects and then 
move them twice as far apart, the force between them will decrease (That's 
the 'inversely' part.) by a factor of FOUR (That's the 'square' part.).

   Newton's Laws can be stated as follows:

   First Law: (Law of Inertia) Objects in motion tend to stay in motion, 
              objects at rest tend to stay at rest, UNLESS ACTED ON BY SOME
              FORCE.
  Second Law: Force equals mass times acceleration, or as an equation F=m*a.
   Third Law: For every action there is an equal and opposite reaction. This
              one shows up most vividly when you try to jump out of a boat. 
              When you jump, the boat pushes you forward with some force, but
              you push the boat backwards with the same force.

   Now, these seem like pretty simple ideas, but they have tremendous 
consequences. I spent two semesters in undergraduate school and another one 
in graduate school, studying the direct consequences of Newton's Laws. They 
are indeed very powerful. Kepler's Laws can be derived mathematically from 
the law of gravitation and Newton's second law in a few pages of 
calculations. The paths of our spacecrafts were calculated using Newton's 
Laws. I can't stress too much how important it is to UNDERSTAND (not just 
memorize) Newton's Laws.

 Dirk
 p.s I see I forgot to put the title in - "WHY they move"
 enjoy!