2cPosition–Time Graphs
Read and interpret x–t graphs: slope equals velocity, curved lines signal acceleration, and every feature of the graph tells a story about motion.
Physicists and engineers constantly read motion graphs. From analyzing crash test data to programming robot movements, the ability to extract velocity from a position–time graph is an essential real-world skill.
Lesson Overview
A position–time (x–t) graph is a powerful visual tool for analyzing motion. In this lesson you will learn to read and interpret x–t graphs: the slope of the line (or tangent) equals velocity, and the shape of the graph reveals whether an object is moving forward, backward, speeding up, slowing down, or standing still.
Key Concepts
Slope of x–t graph
Equals velocity: v = Δx / Δt = rise / run
Positive slope
Object moving in the positive direction (e.g., east or right)
Negative slope
Object moving in the negative direction (e.g., west or left)
Zero slope
Object is stationary — position is not changing
Steeper slope
Greater speed — the object is moving faster
Curved x–t graph
Changing slope means changing velocity — the object is accelerating
An x–t graph shows a straight line from (0 s, 0 m) to (4 s, 20 m). What is the velocity of the object?
An x–t graph shows a straight line from (0 s, 15 m) to (5 s, 0 m). What is the velocity?
An x–t graph has a horizontal line at x = 8 m from t = 2 s to t = 6 s. Describe the motion.
Two objects A and B are shown on the same x–t graph. Object A has a steeper positive slope than object B. What does this tell you about their velocities?
An x–t graph shows a curve that starts with a gentle slope and becomes steeper over time. What type of motion does this represent?
An x–t graph shows a line from (1 s, 4 m) to (5 s, 12 m). Calculate the velocity.
Hint: Velocity = slope = Δx / Δt = (x₂ − x₁) / (t₂ − t₁).
On an x–t graph, what does it mean when two lines cross?
Hint: At the crossing point, both objects are at the same position at the same time.
An x–t graph shows a line with a negative slope. Is the object moving forward or backward? Is it speeding up or slowing down?
Hint: Slope = velocity. Negative slope means negative velocity. A straight line means constant velocity — not speeding up or slowing down.
How would you find the instantaneous velocity at a specific point on a curved x–t graph?
Hint: Draw a tangent line to the curve at that point. The slope of the tangent equals the instantaneous velocity.
An x–t graph shows a line that goes from positive slope to zero slope to negative slope. Describe the motion in words.
Hint: Translate each slope into a velocity description.
Key Vocabulary
Position–Time Graph (x–t graph)
A graph with time on the horizontal axis and position on the vertical axis. The slope equals velocity.
Example: A straight line on an x–t graph indicates constant velocity.
Slope
The ratio of the vertical change (rise) to the horizontal change (run) on a graph: slope = Δy / Δx.
Example: On an x–t graph, slope = Δx / Δt = velocity.
Tangent Line
A straight line that touches a curve at exactly one point and has the same slope as the curve at that point.
Example: The slope of the tangent to an x–t curve at t = 3 s gives the instantaneous velocity at t = 3 s.
Uniform Motion
Motion with constant velocity — represented by a straight line on an x–t graph.
Example: A horizontal line on an x–t graph represents an object at rest (a special case of uniform motion with v = 0).
Interactive Practice — 5 Questions
On a position–time graph, what does the slope represent?
A horizontal line on an x–t graph means the object is:
An x–t graph shows a straight line from (0, 0) to (6 s, −12 m). What is the velocity?
A curved line on an x–t graph indicates:
Two objects have x–t graphs with slopes of +4 m/s and +8 m/s. Which object is moving faster?
Independent Practice
Sketch an x–t graph for the following motion: starts at x = 0, moves to x = 10 m in 5 s, stays at rest for 3 s, then returns to x = 0 in 4 s. Label each segment with its velocity.
An x–t graph shows a straight line from (0 s, 5 m) to (8 s, 29 m). Calculate the velocity and write the equation of motion in the form x = x₀ + vt.
Describe the motion of an object whose x–t graph is a curve that starts steep and becomes less steep (flattens out) over time.
Two cars start at the same position. Car A has a steeper positive slope; Car B has a shallower positive slope. After 10 s, which car is farther from the start? How do you know from the graph?
★ An x–t graph shows a parabola described by x = 2t² (metres, seconds). Find the average velocity between t = 1 s and t = 3 s, and the instantaneous velocity at t = 2 s. Compare the two values and explain any difference.
ChallengeCommon Mistakes
Reading the height (y-value) of the graph as the velocity.
The HEIGHT of the x–t graph gives position. The SLOPE gives velocity. Always calculate Δx / Δt.
Thinking a steep downward slope means the object is slowing down.
A steep negative slope means the object is moving quickly in the negative direction — it is not necessarily decelerating.
Confusing a curved x–t graph with a curved path in space.
An x–t graph shows position vs time, not the physical path. A curve on the graph means changing velocity, not a curved trajectory.
Math Tips
Velocity from x–t graph: v = slope = (x₂ − x₁) / (t₂ − t₁). Always pick two clear points on the line. For a curve, draw a tangent at the point of interest and find its slope.
Equation of motion from a straight x–t graph: x = x₀ + vt, where x₀ is the y-intercept (initial position) and v is the slope.