Before we close our eyes, everyone is collected in front of us. Once our eyes shut our playmates are free to move about the house in search of a good hiding place. When we open our eyes, after counting carefully and steadily to ten, everyone has disappeared! We seek them out, haphazardly at first, but after a few rounds it is discovered that there are only about seven places in the house that people can fit and not be easily seen. To find them we need only check each location they could possibly be, one by one. At this point the excitement has worn off and a new game is invented to be played until everyone must go home for dinner.

The impulse to play hide-and-seek is the same which brought about the discovery of the basic principles of mechanical physics. Balls are thrown into the air and, after a long while, it is discovered that, of all the paths they could take, each ball seems to have a parabolic trajectory. This, now trivial, similarity between the events of balls tossed through the air was identified only after a long, and at times tedious, process of human thought, inquiry, and eventually observation.

Here, we are not worried about the historical order of events which led to our basic understanding of classical kinematics and dynamics, but rather how such ideas can be connected to our everyday experience and imagination. Behind the principles of modern theoretical mechanics, built by Galileo, Kepler, Leibniz, Newton, Euler, Lagrange, and Hamilton (to name a few), are simple similarities between events from everyday life that just so happen to be stated with the greatest amount of clarity and exactness in the language of mathematics.

If you sit in your house looking out a window at two neighborhood kids tossing a ball back and forth you might notice that when they throw the ball really hard they either make a big arc high above their heads or aim the ball right at each other. They never seem to throw the ball softly and straight at each other except when they want it to hit the ground and roll to the other kid.

Being of an acutely curious disposition, you get a marker and trace out the path of the ball on your window. You realize that though you now know what was the shape of the path of the ball, you are unable to say where it happened to be at each moment in time. You get clever, and use string to create a grid over the window so that each part of the window is boxed in by string.

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

You get a metronome and each time it ticks you keep count and write the number of ticks you have heard in the box where the ball is at.

+---+---+---+---+---+---+---+---+---+

| | | | 2 | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | 1 | | 3 | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

| | 0 | | | | 4 | | | |

+---+---+---+---+---+---+---+---+---+

You decide to put this data into a table so that you can make more observations. You quickly realize that you have to label the rows and columns in order to transform your figure into a table.

0 1 2 3 4 5 6 7 8

+---+---+---+---+---+---+---+---+---+

0 | | | | 2 | | | | | |

+---+---+---+---+---+---+---+---+---+

1 | | | 1 | | 3 | | | | |

+---+---+---+---+---+---+---+---+---+

2 | | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

3 | | | | | | | | | |

+---+---+---+---+---+---+---+---+---+

4 | | 0 | | | | 4 | | | |

+---+---+---+---+---+---+---+---+---+

So you make your table with three columns, the first being time (or number of clicks that you've heard from the metronome) the second being the row and the third being the column.

Time Row Column

---------------

0 4 1

1 1 2

2 0 3

3 1 4

4 4 5

You notice a pattern in the numbers: the columns keep going up and the rows go down and back to where they started. This makes sense with what you saw, because the ball went up in the air and then came back down, and it went from one kid to the other. In order to get a better sense of how the location of the ball changed with time you make another table that gives the difference between the location of the ball in a row and column between each click of the metronome.

Intervals Row Change Column Change

------------------------------------

0,1 _3 1

1,2 _1 1

2,3 1 1

3,4 3 1

The change in columns is constant over each time interval, and the change in rows goes from being negative 3 to positive 3 by steps of 2.

Having recently finished your algebra homework, you realize that you know how to write out the functions for these tables, what luck!

Time Row Column

----------------------------

t (t-2)^2 t+1

(t^2)+(_4*t)+4

You perform this same procedure a number of different times and find that each equation you write out for the row and column has a similar form. The column equations always look like (a*t)+b for some numbers a and b, whereas the row equations always look like t^2+(u*t)+v for some numbers u and v. It is odd that EVERY equation for rows has a t^2 term that isn't ever multiplied by any other constant! You are left wanting more, and needing to understand WHY such a strange thing happens...