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6.2Coordinates and Components
‘Cause we’re all
Bold as love,
Just ask the axis.
-Jimi Hendrix
How do we convert these ideas into mathematics. The figure below
shows a good way of connecting the intuitive ideas to the numbers. In one
dimension, we impose a number line with an x coordinate on a certain
stretch of space. In two dimensions, we imagine a grid of squares which we
label with x and y values.
This object experiences a force that pulls it
down toward the bottom of the page. In each
equal time interval, it moves three units to
the right. At the same time, its vertical mo-
tion is making a simple pattern of +1, 0, –1,
–2, –3, –4, ... units. Its motion can be de-
scribed by an x coordinate that has zero ac-
celeration and a y coordinate with constant
acceleration. The arrows labeled x and y
serve to explain that we are defining increas-
ing x to the right and increasing y as upward.
But of course motion doesn’t really occur in a series of discrete hops like
in chess or checkers. The figure on the left shows a way of conceptualizing
the smooth variation of the x and y coordinates. The ball’s shadow on the
wall moves along a line, and we describe its position with a single coordi-
nate, y, its height above the floor. The wall shadow has a constant accelera-
tion of –9.8 m/s
2
. A shadow on the floor, made by a second light source,
also moves along a line, and we describe its motion with an x coordinate,
measured from the wall.
The velocity of the floor shadow is referred to as the x component of the
velocity, written v
x
. Similarly we can notate the acceleration of the floor
shadow as a
x
. Since v
x
is constant, a
x
is zero.
Similarly, the velocity of the wall shadow is called v
y
, its acceleration a
y
.
This example has a
y
=–9.8 m/s
2
.
Because the earth’s gravitational force on the ball is acting along the y
axis, we say that the force has a negative y component, F
y
, but F
x
=F
z
=0.
The general idea is that we imagine two observers, each of whom
perceives the entire universe as if it was flattened down to a single line. The
y-observer, for instance, perceives y, v
y
, and a
y
, and will infer that there is a
force, F
y
, acting downward on the ball. That is, a y component means the
aspect of a physical phenomenon, such as velocity, acceleration, or force,
that is observable to someone who can only see motion along the y axis.
All of this can easily be generalized to three dimensions. In the example
above, there could be a z-observer who only sees motion toward or away
from the back wall of the room.
x
y
x
y
Chapter 6Newton’s Laws in Three Dimensions