**Kinematics** is simply the mathematical description of motion and makes no reference to
the forces which cause the motion. Thus, kinematics is not really part of physics but provides
us with the mathematical framework within which the laws of physics can be stated in a precise
way.

Let us think about a material object (a “particle”) which is constrained to move along a given straight line (e.g., an automobile moving along a straight highway). If we take some point on the line as an origin, the position of the particle at any instant can be specified by a number which gives the distance from the origin to the particle. Positive values of are assigned to points on one side of the origin, and negative values of are assigned to points on the other side of the origin, so that each value of corresponds to a unique point. Which direction is taken as positive and which as negative is purely a matter of convention. The numerical value of clearly depends on the unit of length we are using (e.g., feet, meters, or miles). Unless the particle is at rest, will vary with time. The value of at time is denoted by .

The **average velocity** of a particle during the time interval from to
is defined as

that is, the change in position divided by the change in time. If we draw a graph of versus (e.g., Figure 1) we see that is just the slope of the dashed straight line connecting the points which represent the positions of the particle at times and .

A more important and more subtle notion is that of **instantaneous velocity** (which is
what your car’s speedometer shows). If we hold fixed and let
be closer and closer to , the quotient
will approach a definite limiting
value (provided that the graph of versus is sufficiently
smooth) which is just the slope of the tangent to the versus
curve at the point . This limiting value, which
may be thought of as the average velocity over an infinitesimal time interval which includes
the time , is called the “the instantaneous velocity at time
” or, more briefly, “the velocity at time
” We write

This equation is familiar to anyone who has studied differential calculus; the right side is called “the derivative of with respect to ” and frequently denoted by . Thus .

If is given in the form of an explicit formula, we can calculate
either directly from (2) or by using the rules for
calculating derivatives which are taught in all calculus courses (these rules, e.g.,
, merely summarize the results of
evaluating the right side of (2) for various functional forms of
). A useful exercise is to draw a qualitatively correct graph of
when is given in the form of a graph, rather
than as a formula. Suppose, for example, that the graph of is Figure 2.
We draw a graph of by estimating the **slope** of the
-versus- graph at each point. We see that the slope is
positive at (with a numerical value of about ft/sec,
though we are not interested in very accurate numbers here) and continues positive but with
decreasing values until . The slope is zero between
and and then becomes negative, and so forth (if positive
means that the object is going forward, then negative
means that the object is going backward). An approximate graph of is
given by Figure 3.

If we are given , either as a formula or a graph, we can calculate . The mathematical process of finding the function when its slope is given at all points is called “integration.” For example, if , then where is any constant (the proof is simply to calculate and verify that we obtain the desired ). The appearance of the arbitrary constant in is not surprising, since knowledge of the velocity at all times is not quite sufficient to fully determine the position at all times. We must also know where the particle started, that is, the value of when . If , then .

Suppose we are given the graph of , for example, Figure 4. Let us consider the shaded rectangle whose height is and whose width is , where is a very small time interval.

The area of this rectangle is , which is equal to the
displacement (i.e., the change in ) of the particle during the time interval
from to . (Strictly speaking, the previous
statement is not exactly true unless is constant during the time
interval from to , but if is small enough the variation of during this interval may be
neglected.) If and are any two times, and if
we divide the interval between them into many small intervals, the displacement during any
subinterval is approximately equal to the area of the corresponding rectangle in Figure 5.
Thus the net displacement is approximately equal to the
sum of the areas of the rectangles. If the subintervals are made smaller and smaller, the
error in this approximation becomes negligible, and thus we see that *the area under the
portion of the ** versus ** curve between time ** and ** is equal to the displacement ** experienced by the particle during that time interval.*

The above statement is true even if becomes negative, provided we define the area as negative in regions where is negative. In the notation of integral calculus we write

The right side of (3) is called the “integral of with respect to from to ” and is defined mathematically as the limit of the sum of the areas of the rectangles in Figure 5 as the width of the individual rectangles tends to zero.

Figure 6 shows the velocity of an auto as a function of time. Calculate the distance of the auto from its starting point at , and sec. Calculate the average velocity during the period from sec to sec and during the period from to sec.

Calculating areas: ; ; ; . ; average velocity from to ft/sec; average velocity from to ft/sec (Note: After students have learned more formulas many will use formulas rather than simple calculation of areas and get this wrong.)

A woman is driving between two toll booths miles apart. She drives the first miles at a speed of mph. At what (constant) speed should she drive the remaining miles so that her average speed between the toll booths will be mph?

If is total time, , so hrs. Time for first hr. Therefore, the time for the remaining hr. The speed during the second miles must be mi/hr.

Acceleration is defined as the **rate of change of velocity**. The **average
acceleration** during the interval from to is defined as where