Kinematics in Two Dimensions
synthesis vs. analysis
The geometry that puts algebra on the coordinate plane is called analytic geometry — the topic of the next two sections of this book. Analytic geometry makes a quiet appearance in high school algebra classes. This is when students are introduced to the coordinate plane and when they learn how to graph functions. The topic is rarely identified as analytic geometry, probably because most math teachers don't want to hear the wise guy comment, "This is algebra class. Why are you teaching us geometry?" Analytic geometry is a relatively modern branch of mathematics, going back only as far as the 17th century. It made its way into the secondary math curriculum sometime in the 19th century. Analytic geometry is also known as coordinate geometry, because of the central role played by the coordinate plane, or cartesian geometry, in honor of the French philosopher René Descartes (1596–1650) who proposed the coordinate plane in Discourse on the Method — the book famous for the line "I think, therefore I am".
The geometry that uses logic to test proofs is called synthetic geometry — the topic of this section. Synthetic geometry is a large part of secondary school mathematics. It is usually taught as a full year course in the US. Synthetic geometry was invented in the river civilizations of Egypt, Mesopotamia, and the Indus Valley five thousand years ago. It has been a part of formal education for a long time. The only thing older is arithmetic, which probably predates writing. Synthetic geometry is more commonly known as plane geometry, since the foundations of the subject were laid on two dimensional applications, or euclidean geometry, in honor of the Greek mathematician Euclid of Alexandria (323–283 BCE) who wrote The Elements — the most popular textbook of all time.
Synthesis (joining separated parts into a unified entity) is the opposite of analysis (breaking something up into its constituent parts). Before the 17th century, mathematicians used geometry to solve sets of related problems from which generalized conclusions could be made. After the 17th century, mathematicians applied algebra to generalized problems from which sets of related conclusions could be drawn. Synthesis requires a lot of work to arrive at a few conclusions, while analysis produces a lot of conclusions from a little bit of work. This is a huge oversimplification of the situation, but you get the idea. Mathematics improved dramatically when algebra switched from a tool of geometry to its driving engine.
Synthetic geometry still has its usefulness. I think students react more positively to the "easy" geometry of Euclid's Elements than to the "hard" geometry of Descartes' Discourse. Analytic geometry with its strict adherence to the grid just doesn't give them the warm fuzzy feeling they get when they draw a triangle. Descartes wins out in the end, however. Computers do most of the math in the 21st century and computers love coordinates.
Displacement, velocity, and acceleration (like all vector quantities) are geometric entities. They have magnitude, which is something like length, and direction, which is just the angle they make with respect to a reference direction like "up" or "north". It is always possible to create a vector diagram or a series of diagrams from a properly worded description of a kinematic event. The challenge for those studying this topic is learning how to effectively represent motion with a static picture.
Let's review the kinematic definitions presented in this chapter. First the scalar quantities…
- Distance is a scalar measure of the interval between two locations measured along the actual path connecting them.
- Speed is the rate of change of distance with time.
And then the vector quantities…
- Displacement is a vector measure of the interval between two locations measured along the shortest path connecting them.
- Velocity is the rate of change of displacement with time.
- Acceleration is the rate of change of velocity with time.
Many kinematic problems in Two Dimensions can be solved using synthetic geometry. To solve them you should be able to…
- represent a kinematic event with a geometric diagram
- use synthetic geometry to determine unknown magnitudes (lengths) and directions (angles)
- use these results to determine kinematic quantities
Now let's go and solve some problems.