The Position Function
In the last SparkNote, we discussed position functions in one
dimension. The value of such a function
at a particular time t0, x(t0), was an ordinary number which represented
the position of the object along a single line. In two and three dimensions,
however, the position of an object must be specified by a
vector. We therefore need to upgrade our one-
dimensional function x(t) to x(t), so that at each moment in time
the position of the object is now given in terms of a vector. Whereas x(t)
was a scalar-valued function, x(t) is vector-valued. They
are both, nevertheless, position functions.
As we might expect, the individual components of x(t) correspond to
one-dimensional position functions in each of the two or three directions of
motion. For instance, for motion in three dimensions, the components of
x(t) can be labeled x(t), y(t), and z(t), and correspond to
one-dimensional position functions in the x-, y-, and z-directions,
respectively. If we have three-dimensional motion with constant velocity,
x(t) = vt, where v = (vx, vy, vz) is a
constant vector, the above vector equation for x(t) breaks up into
three one-dimensional equations:
x(t) = vxt, y(t) = vyt, z(t) = vzt
Note that if
vy = vz = 0, what we recover is just one-dimensional motion in the
x-direction.
Position, Velocity, and Acceleration
What makes the generalization to vectors particularly simple is that the
relationships between position, velocity, and acceleration stay exactly the
same. Whereas before we had
v(t) = x'(t) and a(t) = v'(t) = x''(t)
now we have
v(t) = xâ≤(t) and a(t) = vâ≤(t) = xâ≤â≤(t).
where the derivatives are taken
component by component. In other words,
if
x(t) = (x(t), y(t), z(t)), then
xâ≤(t) = (x'(t), y'(t), z'(t)). Therefore, all the equations derived in the previous section are valid
once the scalar-valued functions are turned into vector-valued ones.
As an example, consider the position function
x(
t) =
at2 +
v0t +
x0,
where
a = (0, 0, - g),
v0 = (vx, 0, vz), and
x0 = (0, 0, h). The above vector equation for position can be broken
down into three one-dimensional equations:
x(
t) =
vxt,
y(
t) = 0,
z(
t) = -
gt2 +
vzt +
h
The motion in the
x-direction is of constant velocity, the motion in the
y-direction is non-existent (so really this is a two-dimensional problem), and
the motion in the
z-direction looks like that of an object moving up and down
near the surface of the earth (recall that
g = 9.8 m/s2 is the
acceleration due to gravity near the earth's surface). However, as long as the
vectors
a,
v0, and
x0 are specified, all
of this information can be compressed into a single vector equation.
It is important to keep in mind that, although the vector equations for
kinematics look almost identical to their scalar counterparts, the range of
physical phenomena that they can describe is far greater. The last example
suggests that for the same object, completely different motions can be going on
in the x-, y-, and z-directions, even though they are all part of one
overall motion. This idea of breaking up an object's motion into components
will help us analyze two- and three-dimensional motion by using ideas we've
already learned from the one-dimensional case. In the next
section, we put some of these methods to
work when we discuss motion with constant acceleration in more than one
dimension.
