What Is Which Set of Motion Graphs Is Consistent: A 2024 Guide

When analyzing motion in physics, a key skill is identifying which set of motion graphs is consistent. A consistent set accurately represents the same motion across position-time, velocity-time, and acceleration-time graphs, with no contradictions. These graphs must align through fundamental relationships: the slope of position gives velocity, the slope of velocity gives acceleration, and the area under acceleration gives change in velocity, while the area under velocity gives change in position. Misalignment in shape, sign, or value means the set is inconsistent. Understanding this helps interpret real-world motion, from vehicles on roads to objects in free fall The Physics Classroom.

Understanding the Three Types of Motion Graphs

Motion is described using three graphs: position vs. time, velocity vs. time, and acceleration vs. time. Each reveals different aspects of movement.

The position-time graph shows location over time. A straight line means constant velocity; a curve indicates acceleration. The steeper the slope, the faster the object moves. A horizontal line means the object is at rest.

The velocity-time graph shows speed and direction. A horizontal line means constant velocity; a sloped line means acceleration. The slope of this graph equals acceleration. Positive slope means speeding up in the positive direction; negative slope means slowing down or moving backward. The area under the curve gives displacement Khan Academy.

The acceleration-time graph shows how velocity changes. A flat line above zero means constant positive acceleration; a line at zero means no change in velocity. A negative value means deceleration or acceleration in the negative direction.

For consistency, all three graphs must tell the same story. If the acceleration is zero, velocity must be constant, and position must change linearly. If velocity increases steadily, acceleration must be constant and nonzero, and position must follow a parabolic curve.

Rules for Graph Consistency

Three core rules determine whether a set of motion graphs is consistent.

First, the slope of the position-time graph at any point must equal the velocity at that time. A straight diagonal position line means constant velocity. A curved position graph means velocity is changing—upward curvature means increasing velocity.

Second, the slope of the velocity-time graph must equal acceleration. A straight, sloped velocity line means constant acceleration. A flat velocity line means zero acceleration.

Third, the area under the acceleration-time graph over a time interval must equal the change in velocity. Similarly, the area under the velocity-time graph must match the change in position. For example, if acceleration is 3 m/s² for 4 seconds, velocity must increase by 12 m/s. If the velocity graph shows a smaller change, the set is inconsistent.

Signs must also align. Positive velocity means position increases over time. Negative velocity means position decreases. If velocity is positive but position is decreasing, the graphs conflict. The same applies to acceleration and velocity changes.

Example: Uniform Acceleration

Consider an object starting from rest and accelerating uniformly in the positive direction.

• The position-time graph is a parabola opening upward, showing increasing displacement over time.
• The velocity-time graph is a straight line with positive slope, starting at zero and increasing steadily.
• The acceleration-time graph is a horizontal line above zero, indicating constant positive acceleration.

This set is consistent. The slope of position increases over time, matching the rising velocity. The constant slope of velocity matches the nonzero acceleration. The area under acceleration equals the velocity change. The area under velocity matches the position change.

Now suppose the acceleration graph is zero instead. That would imply constant velocity, but the velocity graph shows increasing speed—contradiction. Or if the position graph were linear while velocity increases, that would also break consistency. Only when all three graphs align through slope and area relationships is the set valid.

How to Check for Consistency Step by Step

To evaluate a set, follow a clear process.

Start with the position-time graph. Is it flat, straight, or curved? If curved, velocity is changing. If straight, velocity is constant. Find the slope at key points—this should match the velocity values.

Next, examine the velocity-time graph. Is it flat or sloped? A constant slope means constant acceleration. The slope should equal the acceleration value at each point. If velocity is increasing linearly, acceleration must be constant and positive.

Then check the acceleration-time graph. If it shows a nonzero constant value, velocity must change steadily and position must curve. If acceleration is zero, velocity must be flat and position linear.

Pick specific times to verify. At t = 3 s, does the slope of the position graph match the velocity shown? Does the slope of velocity match the acceleration value? Even one mismatch means inconsistency.

Use area calculations when needed. If acceleration is 2 m/s² from t = 0 to t = 5 s, velocity should increase by 10 m/s. If the velocity graph only rises by 6 m/s, the graphs do not match. Similarly, if average velocity is 4 m/s over 3 seconds, position should change by 12 meters. Check this against the position graph.

Always confirm units and signs. A velocity graph in m/s paired with a position graph in km will appear off. Negative velocity with increasing position suggests a sign error. Labels and scales matter—unlabeled axes make verification impossible.

Sketching expected graphs from a description also helps. If told “a bike speeds up from rest,” expect a curved position graph, a straight upward-sloping velocity graph, and a flat acceleration line above zero. Compare that to the given set.

Interactive tools can build skill. Khan Academy offers exercises where users match graphs to motions or spot inconsistencies, reinforcing how the graphs relate.

Key Takeaways

• A consistent set of motion graphs shows the same motion across position, velocity, and acceleration without contradictions.
• The slope of the position-time graph equals velocity; the slope of the velocity-time graph equals acceleration.
• The area under the acceleration-time graph equals the change in velocity; the area under the velocity-time graph equals the change in position.
• Common patterns include horizontal lines (rest), straight diagonals (constant velocity), and parabolas (uniform acceleration).
• Always check slopes, areas, signs, and units at specific times to verify alignment.
• Start with one graph, predict the others, and compare step by step to identify which set of motion graphs is consistent.