# Kinematics: The Mathematical Description of Motion

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.

## 1. Motion in One Dimension

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.

### Example 1: Calculating Distance and Average Velocity

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.

### Solution

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.)

### Example 2: Calculating Average Velocity over a Fixed Distance

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?

### Solution

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.

## 2. Acceleration

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