A graph of an object's position over time is a useful tool for analyzing motion. You see such a position-time graph above. Values on the vertical axis represent the mouse car's position, and time is plotted on the horizontal axis. You can see from the graph that the mouse car started at position x = −4 m, then moved to the position x = +4 m at about t = 4.5 s, stayed there for a couple of seconds, and then reached the position x = −2 m again after a total of 12 seconds of motion.

Where the graph is horizontal, as at point B, it indicates the mouse’s position is not changing, which is to say the mouse is not moving. Where the graph is steep, position is changing rapidly with respect to time and the mouse is moving quickly.

Displacement and velocity are mathematically related, and a position-time graph can be used to find the average or instantaneous velocity of an object. The slope of a straight line between any two points of the graph is the object’s average velocity between them.

Why is the average velocity the same as this slope? The slope of a line is calculated by dividing the change in the vertical direction by the change in the horizontal direction, “the rise over the run.” In a position-time graph, the vertical values are the x positions and the horizontal values tell the time. The slope of the line is the change in position, which is displacement, divided by the change in time, which is the elapsed time. This is the definition of average velocity: displacement divided by elapsed time.

You see this relationship stated and illustrated in Equation 1. Since the slope of the line shown in this illustration is positive, the average velocity between the two points on the line is positive. Since the mouse moves to the right between these points, its displacement is positive, which confirms that its average velocity is positive as well.

The slope of the tangent line for any point on a straight-line segment of a position-time graph is constant. When the slope is constant, the velocity is constant. An example of constant velocity is the horizontal section of the graph that includes the point B in the illustration above.

The slope of a tangent line at different points on a curve is not constant. The slope at a single point on a curve is determined by the slope of the tangent line to the curve at that point. You see a tangent line illustrated in Equation 2. The slope as measured by the tangent line equals the instantaneous velocity at the point. The slope of the tangent line in Equation 2 is negative, so the velocity there is negative. At that point, the mouse is moving from right to left. The negative displacement over a short time interval confirms that its velocity is negative.