In order to describe the motion of an object we must be to determine the
position of the object at any point in time. In other words, if we are
given the problem of describing the motion of an object, we will have reached a
solution when we find a position function, x(t), which tells us the
position of that object at any moment in time. (Note that "t" is usually
understood to be a time variable, so in writing the position function
"x" as "x(t)" we are explicitly indicating that position is a
function of time.) There are a variety of functions that can correspond
to the position of moving objects. In this section we will introduce some of
the more common ones that tend to arise in basic physics problems.
Examples of Position Functions
x(t) = c, where c is a constant. As you might expect, an object that has
this as its position function isn't going anywhere. At all times its position
is exactly the same: c.
x(t) = vt + c, where v and c are constants. An object with this position
function starts off (at t = 0) with a position c, but its position changes
with time. At a later time, say t = 5, the object's new position will be given
by x(5) = 5v + c. Because the exponent of t in the above equation is 1, we
say the object changes linearly with time. Such objects are moving at a
constant velocity (which is why the coefficient of "t" has been
suggestively labeled v).
x(t) = 1/2at2, where a is a constant. At t = 0, this object is
situated at the origin, but its position changes quadratically with time
(since the exponent of t in the above equation is 2). For positive a, the
graph of this position function looks like a parabola that touches the
horizontal axis (the time-axis) at the point t = 0. For negative values of a,
the graph of this function is an upside-down parabola. Such a position function
corresponds to objects undergoing constant acceleration (which is why the
coefficient of "t2" has been conveniently written as 1/2a).
x(t) = cos wt, where w is a constant. An object with this position
function is undergoing simple harmonic motion, which means its position is
oscillating back and forth in a special fashion. Since the range of the cosine
function is (- 1, 1), the object is constrained to move within this small
interval and will forever be retracing its path. An example of such an object
is a ball hanging from a spring that is bouncing up and down. In contrast to
the above three examples, this kind of function describes motion where neither
the position, velocity, nor acceleration of the object are constant.
It is probably clear by now that, although the position function of an object is
our ultimate goal in solving kinematics problems, position is closely related to
other quantities such as velocity and acceleration. In the next
section we will make such
relationships more precise, and find that knowledge of the velocity or
acceleration of an object can help us find its position function. Conversely,
knowledge of an object's position function is all we need to reconstruct its
velocity and acceleration functions.