Motion Graphs: Position, Velocity, & Acceleration

Motion graphs, also known as kinematic curves, are a common way to diagram the motion of objects in physics. The three graphs of motion a high school physics student needs to know are: the position vs. time graph (x vs. t), velocity vs. time graph (v vs. t), and acceleration vs. time graph (a vs. t).

Each of these graphs helps to tell the story of the motion of an object. Moreover, when the position, velocity, and acceleration of an object are graphed over the same time interval, the shapes of each graph relate in a specific and predictable way. The principles behind these graphs and the relationships between position, velocity, and acceleration are very likely to show up on the AP physics tests and many other areas of physics.

The x-axis on all motion graphs is always time, measured in seconds. The axis is thus always labeled t(s).

The y-axis on each graph is position in meters, labeled x(m); velocity in meters per second, labeled v(m/s); or acceleration in meters per second squared, labeled a(m/s²)

Motion graphs are often (though certainly not always) sketched without graphing specific points, instead showing a general shape that describes the relative motion of an object.

The position of an object can be positive or negative, depending on the frame of reference. Whatever the diagram shows, the coordinate plane must match. Consider the example of a kid riding along a straight line east and then west on her bike. Call east the positive direction and west the negative direction.

For the first five seconds of her ride (from t = 0 to t = 5), she was moving at a constant rate to the east. This is indicated by a straight, increasing line in the positive quadrant of the position-time graph. Another way to think of it is that her position is increasing positively. In the next three seconds (t = 5 to t = 8), she stopped for a break. Her position does not change in this time period, indicated by a constant horizontal line stuck at +10 meters.

Finally, the girl on the bike in the last part of her ride (t = 8 to t = 15) begins accelerating back in the westward direction. This is indicated by a line that is non-constant (curved) and heading into the negative quadrant of the graph, meaning her position decreases relative to her starting parameters. The slope of the line increases over time, in the negative direction, showing that her speed is increasing as she covers more ground every second.

Position-time graphs lead directly to velocity-time graphs: The slope of a position-time line shows the velocity of the object in the same time interval. This makes sense because position vs. time is just another way of saying meters per second — velocity. In this case, the only difference is what goes on the y-axis. Consider the same girl on her bike. For the first five seconds of her ride, she traveled 10 meters in five seconds, or 2 meters per second. To graph her velocity in that same time interval then, find 2 m/s on the y-axis and draw a flat line for the first five seconds. Remember, she is biking at a constant velocity, so the slope on this graph is zero. She is also moving with a positive velocity.

Then, for the next three seconds, she didn't move at all, so her velocity abruptly dropped to zero. (Realistically, yes, she must have decelerated from 2 m/s to 0 m/s in more than an instant. But for the sake of simplicity here, consider that her speed changed instantaneously.) Of course, if her velocity is zero, that means there should be no curve on the graph over that time interval. In other words, the curve is now directly on top of the x-axis. Finally, the girl started picking up speed, backtracking homeward. Here the velocity graph gets interesting because there is a changing velocity (meaning there is a non-zero acceleration). Assuming she had a constant acceleration — that is, each second she was increasing her velocity by the same amount as the second before — this means her velocity was increasing at a constant rate. The twist in this scenario is that she also changed direction.

Altogether, that means the velocity-time graph for the last segment of her ride (t = 8 to t = 15) needs to show a straight line where her velocity is growing negatively. In other words, a straight line moving from the x-axis at t = 8 seconds in a diagonal towards the bottom right of the graph.

The acceleration of the object in question can often be tricky for students; just remember the meaning of acceleration: a change in velocity.

For the first eight seconds of her ride, the girl's velocity was not changing. (Again, ignoring her instantaneous shift from 2 m/s to stopped.)

That means for the first eight seconds, her acceleration was zero.

Making the motion graph for this, where the y-axis is now showing acceleration in m/s2, is therefore pretty simple (the horizontal axis is still time):

Now, for the last portion of her ride, recall that her velocity was increasing at a constant rate in the negative direction. Since increasing velocity is acceleration, the acceleration-time graph should have a horizontal line in the negative quadrant from eight seconds onward.

In the real world, acceleration is frequently not constant. On the acceleration-time graph, this would look like a curved line. Calculating the corresponding position-time and velocity-time graphs to go with this is typically beyond the scope of a non-calculus-based physics course. Students are expected to realize a curved line is not constant, however, and that the graph indicates a changing acceleration.

The graph for position should look roughly similar to what it did before, but with any sharp corners smoothed out. The same would happen with the velocity graph — rough corners become smoothed out. But in addition, the instantaneous jump on the velocity graph from 2 m/s to 0 m/s becomes a smooth, slanted line with a large negative slope instead of a vertical line.

On the acceleration graph, at around the five second mark, a steep curve would dip into the negative region before coming back to 0 to indicate the negative acceleration required to come to a stop. And the jump that occurs at the 8-second mark, instead of being vertical, would simply curve into a line with a large negative slope that then smoothly flattens out at -0.5 m/s/s.

When an object moves with some speed and acceleration from its initial position, we can use these three graphs and the measurements they represent to describe most of its behavior. There are a couple useful tricks to remember these relationships.

The slope of the position vs. time graph across an interval is the velocity. The slope of the velocity vs. acceleration graph across an interval is acceleration. A positive slope or a negative slope indicate the direction of velocity or acceleration.

References

• The Physics Classroom: The Meaning of Shape for a P-T Graph
• The Physics Hypertextbook: Graphs of Motion