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 position versus time graphs:

The first graph gives 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.

In contrast, object B is only changing its position by 1 meter every second. Thus, it is moving with a speed of 1 m/s.

Finally, 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.

Intercept and Initial Position

Assuming the motion is defined to begin at t = 0, then the initial position of the motion (the position at time t = 0) can be found by taking the y-intercept of the position versus time graph. Thus, in the examples above, objects A and C begin thier motions at position x = 0, while object B begins its motion at position x = 5 m.

Graphical Representation of Constant Acceleration

Parabolic Form

From the formulas given in the model One-Dimensional Motion with Constant Acceleration, it is clear that a plot of position vs. time will give a parabola.

If the acceleration is positive the parabola will open upwards. The position at t = t_i will be x_i , as shown in the graph below (time at the origin is t_i ):

In this case the position is positive. The fact that the plot of position vs. time is increased means that the initial velocity, v_i , is also positive.

If the acceleration and the initial position x_i were the same, but the initial velocity was negative , then the graph of position vs. time would look like this:

The parabola has a minimum value at the time t_min

Concavity and Acceleration

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.

"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 that Represent Speeding Up

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 that Represent Slowing Down

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.)