# Kinematics

## Graphical Analysis of One-Dimensional Motion


### Learning Objectives

By the end of this section, you will be able to:
1. Describe a straight-line graph in terms of its slope and y-intercept.
2. Determine average velocity or instantaneous velocity from a graph of position vs. time.
3. Determine average or instantaneous acceleration from a graph of velocity vs. time.
4. Derive a graph of velocity vs. time from a graph of position vs. time.
5. Derive a graph of acceleration vs. time from a graph of velocity vs. time.
A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of position, velocity, and acceleration versus time to illustrate one-dimensional kinematics.

### Slopes and General Relationships

First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. If we call the horizontal axis the -axis and the vertical axis the -axis, as in , a straight-line graph has the general form
Here  is the slope, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter  is used for the , which is the point at which the line crosses the vertical axis.

### Graph of Position vs. Time (a = 0, so v is constant)

Time is usually an independent variable that other quantities, such as position, depend upon. A graph of position versus time would, thus, have  on the vertical axis and  on the horizontal axis.    is just such a straight-line graph. It shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.
Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity  and the intercept is position at time zero—that is, . Substituting these symbols into  gives
or
Thus a graph of position versus time gives a general relationship among displacement(change in position), velocity, and time, as well as giving detailed numerical information about a specific situation. 
From the figure we can see that the car has a position of 525 m at 0.50 s and 2000 m at 6.40 s. Its position at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.

### Graphs of Motion when  is constant but 

The graphs in  below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the position and velocity are initially 200 m and 15 m/s, respectively.
The graph of position versus time in (a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in (a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in (b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in (c).
Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a  vs.  graph, rise = change in velocity  and run = change in time .
Since the velocity versus time graph in (b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in (c).
Additional general information can be obtained from  and the expression for a straight line, .
In this case, the vertical axis  is , the intercept  is , the slope  is , and the horizontal axis  is . Substituting these symbols yields
A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension.

 
It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to discover physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.

### Graphs of Motion Where Acceleration is Not Constant

Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in . Time again starts at zero, and the initial velocity is 165 m/s. (This was the final velocity of the car in the motion graphed in .) Acceleration gradually decreases from  to zero when the car hits 250 m/s. The velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.
A graph of position versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point. Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.

### Section Summary

1. Graphs of motion can be used to analyze motion. 
2. Graphical solutions yield identical solutions to mathematical methods for deriving motion equations.
3. The slope of a graph of displacement  vs. time  is velocity .
4. The slope of a graph of velocity  vs. time  graph is acceleration .
5. Average velocity, instantaneous velocity, and acceleration can all be obtained by analyzing graphs.

### Conceptual Questions


### Problems & Exercises

Note: There is always uncertainty in numbers taken from graphs. If your answers differ from expected values, examine them to see if they are within data extraction uncertainties estimated by you.