By Amy Dusto
Updated: March 14, 2025 7:39 pm EST

Sean Gladwell/Getty Images

Motion graphs—also known as kinematic curves—are essential tools for visualizing how objects move. In a high‑school physics curriculum, students encounter three core graphs: position versus time (x vs. t), velocity versus time (v vs. t), and acceleration versus time (a vs. t). These plots not only illustrate the motion of a single object but also reveal the interdependent nature of position, velocity, and acceleration. Understanding these relationships is crucial for AP Physics exams and many physics applications.

Setting Up Motion Graphs

The horizontal axis on every motion graph represents time, labeled t (s). The vertical axis depends on the quantity plotted: position (x, m), velocity (v, m/s), or acceleration (a, m/s²). While exact points can be plotted, many introductory lessons favor sketching a general shape that captures the qualitative behavior of the motion.

Position‑Time Graphs

Position can be positive or negative based on the chosen reference direction. For example, if a cyclist rides eastward (positive) and later westward (negative), the graph’s quadrant reflects this choice. In a typical scenario:

• From t = 0 s to 5 s: The cyclist moves east at a constant speed, producing a straight, positively sloped line in the positive quadrant.
• From t = 5 s to 8 s: She stops; the graph shows a horizontal line at +10 m, indicating no change in position.
• From t = 8 s to 15 s: She accelerates westward, generating a concave‑down curve that extends into the negative quadrant. The increasing negative slope reflects a growing magnitude of westward speed.

Velocity‑Time Graphs

The slope of a position‑time curve directly yields velocity. In the cyclist example:

• 0 s–5 s: Constant velocity of 2 m/s (10 m / 5 s). Draw a horizontal line at +2 m/s.
• 5 s–8 s: Zero velocity; the graph lies on the t‑axis.
• 8 s–15 s: Velocity decreases linearly into the negative domain. A straight line sloping downward from the origin of this interval to –3 m/s at 15 s reflects constant negative acceleration.

Acceleration‑Time Graphs

Acceleration is the rate of change of velocity. For the cyclist:

• 0 s–8 s: Velocity remains constant, so acceleration is zero—draw a horizontal line at 0 m/s².
• 8 s–15 s: Velocity changes at a constant negative rate; the graph shows a horizontal line at –0.5 m/s² (or whatever the calculated value is).

More Realistic Motion Maps

In practice, acceleration often varies, producing curved lines on the acceleration‑time plot. Correspondingly, the velocity‑time graph will have a curved slope, and the position‑time graph will be smoother, with no sharp corners. The abrupt jumps in velocity (e.g., 2 m/s to 0 m/s) become gradual transitions reflecting finite deceleration.

General Relationships to Remember

The three graphs are mathematically linked:

• Velocity is the derivative of position: v = dx/dt. Therefore, the slope of the position‑time graph equals velocity.
• Acceleration is the derivative of velocity: a = dv/dt. Thus, the slope of the velocity‑time graph equals acceleration.
• Conversely, the area under the velocity‑time curve equals the change in position, and the area under the acceleration‑time curve equals the change in velocity.

References

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