a/The car can change its x
and y motions by one square
every turn.3.4 Motion in three dimensions
3.4.1 The Cartesian perspective
When my friends and I were bored in high school, we used to play
a paper-and-pencil game which, although we never knew it, was Very
Educational — in fact, it pretty much embodies the entire world-
view of classical physics. To play the game, you draw a racetrack on
graph paper, and try to get your car around the track before anyone
else. The default is for your car to continue at constant speed in a
straight line, so if it moved three squares to the right and one square
up on your last turn, it will do the same this turn. You can also
control the car’s motion by changing its ∆xand ∆yby up to one
unit. If it moved three squares to the right last turn, you can have
it move anywhere from two to four squares to the right this turn.b/French mathematician Rene Descartes invented analytic geometry; ́
Cartesian (x y z) coordinates are named after him. He did work in phi-
losophy, and was particularly interested in the mind-body problem. He
was a skeptic and an antiaristotelian, and, probably for fear of religious
persecution, spent his adult life in the Netherlands, where he fathered
a daughter with a Protestant peasant whom he could not marry. He kept
his daughter’s existence secret from his enemies in France to avoid giving
them ammunition, but he was crushed when she died of scarlatina at age- A pious Catholic, he was widely expected to be sainted. His body was
buried in Sweden but then reburied several times in France, and along
the way everything but a few fingerbones was stolen by peasants who
expected the body parts to become holy relics.
The fundamental way of dealing with the direction of an ob-
ject’s motion in physics is to use conservation of momentum, since
momentum depends on direction. Up until now, we’ve only done
momentum in one dimension. How does this relate to the racetrack
game? In the game, the motion of a car from one turn to the next
is represented by its ∆xand ∆y. In one dimension, we would only
need ∆x, which could be related to the velocity, ∆x/∆t, and the
momentum,m∆x/∆t. In two dimensions, the rules of the game
amount to a statement that if there is no momentum transfer, then
bothm∆x/∆tandm∆y/∆tstay the same. In other words, there
are two flavors of momentum, and they areseparately conserved.
All of this so far has been done with an artificial division of time
into “turns,” but we can fix that by redefining everything in terms
of derivatives, and for motion in three dimensions rather than two,
we augmentxandywithz:
vx= dx/dt vy= dy/dt vz= dz/dt
and
px=mvx py=mvy pz=mvzSection 3.4 Motion in three dimensions 191