We have already discussed examples of position functions in the previous section. We now turn our attention to velocity and acceleration functions in order to understand the role that these quantities play in describing the motion of objects. We will find that position, velocity, and acceleration are all tightly interconnected notions.

In one dimension, *velocity* is almost exactly the same as what we normally
call *speed.* The speed of an object (relative to some fixed
reference frame) is a measure of "how fast" the object is going--and
coincides precisely with the idea of speed that we normally use in reference to
a moving vehicle. Velocity in one-dimension takes into account one additional
piece of information that speed, however, does not: the *direction* of the
moving object. Once a coordinate axis has been chosen for a particular problem,
the *velocity*
*v*
of an object moving at a speed
*s*
will either be
*v* = *s*
,
if the object is moving in the positive direction, or
*v* = - *s*
, if the object is
moving in the opposite (negative) direction.

More explicitly, *the velocity of an object is its change in position per
unit time,* and is hence usually given in units such as m/s (meters per
second) or km/hr (kilometers per hour). The velocity function,
*v*(*t*)
, of an
object will give the object's velocity at each instant in time--just as the
speedometer of a car allows the driver to see how fast he is going. The value
of the function
*v*
at a particular time
*t*
_{0}
is also known as the
instantaneous velocity of the object at time
*t* = *t*
_{0}
, although the word
"instantaneous" here is a bit redundant and is usually used only to emphasize
the distinction between the velocity of an object at a *particular instant*
and its "average velocity" over a longer time interval. (Those familiar with
elementary calculus will recognize the velocity function as the *time
derivative* of the position function.)

Now that we have a better grasp of what velocity is, we can more precisely define its relationship to position.

We begin by writing down the formula for average velocity. The average
velocity of an object with position function
*x*(*t*)
over the time interval
(*t*
_{0}, *t*
_{1})
is given by:

In other words, the average velocity is the total displacement divided by the total time. Notice that if a car leaves its garage in the morning, drives all around town throughout the day, and ends up right back in the same garage at night, its displacement is 0, meaning its average velocity for the whole day is also 0.

As the time intervals get smaller and smaller in the equation for average
velocity, we approach the instantaneous velocity of an object. The formula we
arrive at for the velocity of an object with position function
*x*(*t*)
at a
particular instant of time
*t*
is thus:

This is, in fact, the formula for the velocity function in terms of the position function! (In the language of calculus, this is also known as the formula for the

Loosely speaking, the time derivative of a function
*f* (*t*)
is a new function
*f'*(*t*)
that keeps track of the rate of change of
*f*
in time. Just as in our
formula for velocity, we have, in general:

Notice that this means we can write:

We will see later that this enables us to write:

It can be shown, from the above definition for the derivative, that derivatives satisfy certain properties:

- (P1)
(
*f*+*g*)' =*f'*+*g'* - (P2)
(
*cf*)' =*cf'*, where*c*is a constant.

- (F1) if
*f*(*t*) =*t*^{n}, where*n*is a non-zero integer, then*f'*(*t*) =*nt*^{n-1}. - (F2) if
*f*(*t*) =*c*, where*c*is a constant, then*f'*(*t*) = 0 . - (F3a)if
*f*(*t*) = cos*wt*, where*w*is a constant, then*f'*(*t*) = -*w*sin*wt*. - (F3b)if
*f*(*t*) = sin*wt*, then*f'*(*t*) =*w*cos*wt*.

Since we know that
*v*(*t*) = *x'*(*t*)
, we can now use our new knowledge of derivatives
to compute the velocities for some basic position functions:

- for
*x*(*t*) =*c*,*c*a constant,*v*(*t*) = 0 (using (F2)) - for
*x*(*t*) =*at*^{2}+*vt*+*c*,*v*(*t*) =*at*+*v*(using (F1),(F2),(P1), and (P2)) - for
*x*(*t*) = cos*wt*,*v*(*t*) = -*w*sin*wt*(using (F3a)) - for
*x*(*t*) =*vt*+*c*,*v*(*t*) =*v*(using (F1),(P2))

Just as velocity is given by the *change in position per unit time,*
acceleration is defined as the *change in velocity per unit time,* and is
hence usually given in units such as m/s
^{2}
(meters per second
^{2}
; do not be
bothered by what a second
^{2}
is, since these units are to be interpreted as
(m/s)/s--i.e. units of velocity per second.) From our past experience with the
velocity function, we can now immediately write by analogy:
*a*(*t*) = *v'*(*t*)
, where
*a*
is the acceleration function and
*v*
is the velocity function. Recalling
that
*v*
, in turn, is the time derivative of the position function
*x*
, we find
that
*a*(*t*) = *x''*(*t*)
.

To compute the acceleration functions corresponding to different velocity or position functions, we repeat the same process illustrated above for finding velocity. For instance, in the case

we find

Combining this latest result with (2) above, we discover that, for constant
acceleration
*a*
, initial velocity
*v*
_{0}
, and initial position
*x*
_{0}
,

This position function represents

A natural question to ask might be, "Why stop at acceleration? If
*v*(*t*) = *x'*(*t*)
,
and
*a*(*t*) = *x''*(*t*)
, why don't we discuss
*x'''*(*t*)
and so forth?" It turns out,
the third time derivative of position,
*x'''*(*t*)
, *does* have a name: it
is called the "jerk" (honestly). The nice thing is, however, that these higher
derivatives don't seem to come into play in formulating physical laws. They
exist and we can compute them, but when it comes to writing down force laws
(such as Newton's Laws) which deal with the
*dynamics* of physical systems, they get completely left out. This is why
we don't care so much about giving them special names and computing them
explicitly.