One-Dimensional Motion and Graphs

Graphical Analysis of One-Dimensional Motion

By the end of this lesson and the next few, you should be able to:

• Describe a straight-line graph in terms of its slope and y-intercept.
• Determine average velocity or instantaneous velocity from a graph of position vs. time.
• Determine average or instantaneous acceleration from a graph of velocity vs. time.
• Derive a graph of velocity vs. time from a graph of position vs. time.
• 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 x-axis and the vertical axis the y-axis, as in the figure below, a straight-line graph has the general form

\(y = mx + b.\)

Here \(m\) is the slope, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter \(b\) is used for the y-intercept, which is the point at which the line crosses the vertical axis.

A straight-line graph. The equation for a straight line is \(y = mx + b\). Image credit: OpenStax, College Physics