A set of motion graphs is consistent when the position-time, velocity-time, and acceleration-time graphs all describe the same motion using three connected rules: the slope of the position graph equals the velocity, the slope of the velocity graph equals the acceleration, and the area under the velocity graph equals the change in position. If any of those relationships break down between the graphs, the set is inconsistent.
This question appears frequently on physics exams, and it tests whether you can read all three graphs as a linked system rather than treating each one in isolation. Here’s how to check any set of graphs quickly and reliably.
The Three Rules That Link Motion Graphs
Every consistent set of motion graphs obeys the same chain of relationships. The slope of the position-time graph at any moment gives the velocity at that moment. The slope of the velocity-time graph at any moment gives the acceleration at that moment. And working in the other direction, the area under the acceleration-time graph gives the change in velocity, while the area under the velocity-time graph gives the change in position (displacement).
Think of it as a hierarchy. Acceleration is the most “basic” graph, velocity is built from it, and position is built from velocity. If you know the acceleration graph and the starting velocity, you can reconstruct the velocity graph. If you know the velocity graph and the starting position, you can reconstruct the position graph. A consistent set is one where all three graphs could actually be reconstructed from each other this way.
What Each Graph Shape Tells You
Constant acceleration is the scenario you’ll encounter most often. When acceleration is constant (a horizontal line on the acceleration-time graph), the velocity-time graph is a straight line with a slope equal to that acceleration value. The position-time graph is a parabola. If you see a set of graphs showing constant acceleration, a curved velocity graph, and a straight-line position graph, that set is inconsistent because the shapes don’t follow the rules.
Here’s a quick reference for the most common motion types:
- Object at rest: Position graph is a horizontal line. Velocity graph sits at zero. Acceleration graph sits at zero.
- Constant velocity (no acceleration): Position graph is a straight line with a nonzero slope. Velocity graph is a horizontal line at some value above or below zero. Acceleration graph sits at zero.
- Constant acceleration: Position graph is a parabola. Velocity graph is a straight line with a nonzero slope. Acceleration graph is a horizontal line at some nonzero value.
- Changing acceleration: Position graph is a more complex curve. Velocity graph is curved. Acceleration graph is not horizontal.
How to Check Consistency Step by Step
Start with the acceleration-time graph because it’s usually the simplest. Read its value at a given time. That value should match the slope of the velocity-time graph at the same time. If the acceleration graph shows a constant positive value of, say, 2 m/s², the velocity graph must be a straight line tilting upward with a rise of 2 m/s for every 1 second.
Next, check the velocity-time graph against the position-time graph. Wherever velocity is positive, position should be increasing. Wherever velocity is zero, the position graph should be momentarily flat (its slope is zero at that instant). Wherever velocity is negative, position should be decreasing. If the velocity graph shows a constant positive value and the position graph is curving or decreasing, something is wrong.
Finally, check the curvature of the position graph. When acceleration is positive, the position-time graph curves upward (concave up, like a bowl). When acceleration is negative, it curves downward (concave down, like an upside-down bowl). When acceleration is zero, the position graph has no curvature at all; it’s a straight line.
The Most Common Mistakes to Watch For
Research on physics students consistently finds one dominant error: confusing the height of a graph with its slope. For example, students see that the position graph is high on the y-axis and assume the velocity must also be high. But velocity depends on the slope of the position graph, not its height. An object can be far from the starting point (high position value) while moving slowly or even standing still, as long as the position graph is nearly flat at that moment.
Another common trap involves negative acceleration. Many students assume that negative acceleration always means the object is slowing down. That’s only true when the object is moving in the positive direction. If an object moves in the negative direction (velocity is negative) and acceleration is also negative, the object is actually speeding up. When checking graph consistency, pay attention to the signs of both velocity and acceleration rather than assuming negative acceleration means deceleration.
A third mistake is expecting the position graph to look like the velocity graph. They describe different quantities and generally have different shapes. In constant acceleration, the velocity graph is a straight line while the position graph is a parabola. They should not look similar.
A Worked Example
Imagine you’re given three graphs. The acceleration-time graph shows a constant value of +3 m/s² for 4 seconds. The velocity-time graph is a straight line starting at 0 m/s and rising to 12 m/s over those 4 seconds. The position-time graph is a parabola starting at 0 and curving upward.
Check 1: The slope of the velocity graph is (12 − 0) / (4 − 0) = 3 m/s². That matches the acceleration graph. Good.
Check 2: The area under the velocity graph is the area of a triangle with base 4 s and height 12 m/s, which equals ½ × 4 × 12 = 24 m. So the position graph should end at 24 m. If it does, the graphs are consistent.
Check 3: The position graph is concave up, which matches the positive acceleration. Velocity is always positive during this interval, and position is always increasing. Everything lines up.
Now imagine the same acceleration and velocity graphs, but the position graph is a straight line from 0 to 24 m. That set is inconsistent. A straight position-time line means constant velocity, but the velocity graph clearly shows the speed changing. The parabolic shape is required.
Quick Consistency Checklist
When you’re staring at a multiple-choice problem, run through these checks in order:
- Slope of position graph = value on velocity graph at every time point. Steep position graph means large velocity. Flat position graph means zero velocity.
- Slope of velocity graph = value on acceleration graph at every time point. Steep velocity graph means large acceleration. Flat velocity graph means zero acceleration.
- Signs must agree. Positive velocity means position is increasing. Positive acceleration means velocity is increasing.
- Curvature must agree. Positive acceleration means the position graph curves upward. Negative acceleration means it curves downward. Zero acceleration means the position graph is straight.
- Areas must agree. The area under the velocity curve over any time interval equals the change in position during that interval.
If all five checks pass, the set of graphs is consistent. If any single check fails, eliminate that option and move on.