Position Versus Time Graph

A plot of position as a function of time is an often useful diagrammatic representation of kinematics problems.

Graphical Representation of Constant Velocity

Slope and Velocity

The mathematical definition of velocity is equivalent to the formula for the slope of a position versus time graph. To see the utility of this correspondence, consider the following plots:

null null null

Consider the left plot. The abscissa is time, and the ordinate is position. This graph is giving the position of some object (called object A) as a function of time. Looking at the graph, we can see that for each second of time that elapses, the object changes its position by 2 meters. This is the same as saying that the slope of the left plot is 2 m / (1 s) or, more simply, 2 m/s. Object A, then, is moving with a speed of 2 m/s.

Contrast that with the middle plot. Object B is only changing its position by 1 meter every second. Thus, it is moving with a speed of 1 m/s.

Finally, look at the plot on the right. Object C is changing its position by 2 meters every second, and so it has a speed of 2 m/s. Note that objects A and C have the same speed. The graphs are different, however, because object C is moving in the negative direction.

Graphical Representation of Constant Acceleration

Concavity and Acceleration

Because of the similarity of the definitions of acceleration and velocity, acceleration can be thought of as the slope of a [velocity versus time graph], just as velocity is the slope of a position versus time graph. It might not be clear, however, that we can also see the effects of acceleration in a position versus time graph.

null

Consider the position vs. time graph shown above. If you were to lay a ruler along the curve of the graph at the origin, the ruler would have to be horizontal to follow the curve, indicating zero slope. Thus, the velocity is zero at the origin. As you follow the curve, however, the ruler would have to be held at a steeper and steeper angle (see the lines added in the graph below). The slope grows with time, indicating that the velocity is becoming more and more positive (the speed is increasing). This positive change in velocity indicates a positive acceleration. In calculus terminology, we would say that a graph which is "concave up" or has positive curvature indicates a positive acceleration.

null

"Acceleration" versus "Deceleration"

In everyday speech, we distinguish between "accelerating" (speeding up) and "decelerating" (slowing down). In physics, both situations are referred to as acceleration (which can be confusing). It is possible to give an exact definition of deceleration, however. Deceleration occurs when the velocity and the acceleration vectors have opposite directions. "Acceleration" in the everyday sense (speeding up) occurs when the acceleration vector and the velocity vector have the same direction. The two cases can be distinguished graphically.

Graphs Showing "Acceleration"

Both the graphs that show "acceleration" in the everyday sense (speeding up) have slopes that are steepening with time. The only difference is that one of the graphs has a steepening positive slope and the other has a steepening negative slope.

Graphs Showing "Deceleration"

Both graphs showing "deceleration" (slowing down) have slopes that are approaching zero as time evolves. (Again, one has a negative slope and one has a positive slope.)