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Motion Graphs

Introduction

Motion graphs turn movement into a visual story. Instead of describing every second of a journey in words, you can place time on the horizontal axis and a motion quantity such as position, distance, speed, velocity, or acceleration on the vertical axis. The shape, slope, and area of a graph can then reveal how an object moves.

This aiMOOC is designed for Grades 9–10. You will learn how to read and create motion graphs, calculate motion quantities from them, translate between different graph types, and explain what a graph means in a real situation. The course links kinematics, graphs, slope, Speed, Velocity, Acceleration, and displacement.

A distance–time graph is a useful starting point because its steepness shows how quickly distance changes with time.

The video above shows how the same motion can be represented with tables, diagrams, and position–time graphs.

Learning Goals

By the end of this course, you should be able to explain the difference between distance, displacement, speed, velocity, and acceleration; identify the variables and units on a motion graph; calculate slope correctly; interpret positive, zero, and negative slopes; use area under speed–time and velocity–time graphs; compare position–time, velocity–time, and acceleration–time representations; create graphs from measurements or descriptions; and justify conclusions with evidence from a graph.

Foundations of Motion

Scalars and Vectors

Distance is the total length of the path traveled. It is a scalar, so it has magnitude but no direction. Displacement is the change in position from the starting point to the ending point. It is a vector, so direction matters.

Speed tells you how fast distance changes. Velocity tells you how fast displacement changes and includes direction. For average values over a time interval,

$average speed = \frac{distance traveled}{Δ t}$

and

$average velocity = \frac{Δ x}{Δ t}$ .

If a runner travels 100 m east and then 100 m west back to the starting point, the total distance is 200 m but the displacement is 0 m.

Axes, Scales, and Units

On most motion graphs, time is on the horizontal x-axis. The vertical y-axis shows the quantity being studied. Always read the axis labels and units before interpreting the graph.

A position–time graph may use meters and seconds. A velocity–time graph may use meters per second and seconds. An acceleration–time graph may use meters per second squared and seconds. A numerical slope is meaningful only when you include the units produced by dividing the vertical-axis unit by the horizontal-axis unit.

Position–Time and Distance–Time Graphs

Reading Position–Time Graphs

A position–time graph shows an object's location relative to an origin as time passes. The slope tells you velocity:

$v = \frac{Δ x}{Δ t}$ .

A positive slope means positive velocity. A horizontal segment means the object is at rest because its position does not change. A negative slope means the object is moving in the negative direction. A steeper slope means a velocity with greater magnitude.

For a straight segment, the slope is constant, so the velocity is constant. For a curve, the slope changes. You can estimate instantaneous velocity by finding the slope of a tangent line at the point of interest.

Distance–Time Graphs Need Careful Language

A true total-distance–time graph cannot slope downward because total path length does not decrease as a journey continues. Its slope gives speed. A horizontal segment means no additional distance is being traveled.

Some school graphs labeled “distance–time” actually show distance from a reference point. Such a graph can slope downward when the object returns toward the reference point. In that situation, treat the vertical coordinate more like a one-dimensional position magnitude and read the graph description carefully.

Worked Example: Position–Time Slope

Suppose a cyclist's position changes from 10 m to 34 m between 2 s and 8 s. The change in position is 24 m and the elapsed time is 6 s. The average velocity is

$v = \frac{34 - 10}{8 - 2} = 4 m/s$ .

If the graph is a straight line over this interval, the cyclist's velocity is constant at 4 m/s.

Velocity–Time and Speed–Time Graphs

Slope Means Acceleration

On a velocity–time graph, slope represents acceleration:

$a = \frac{Δ v}{Δ t}$ .

A horizontal line means constant velocity and therefore zero acceleration. An upward slope means positive acceleration. A downward slope means negative acceleration. Whether the object is speeding up or slowing down depends on both the sign of velocity and the sign of acceleration.

Area Means Displacement or Distance

The signed area between a velocity–time graph and the time axis gives displacement. Area above the time axis is positive; area below it is negative. For a rectangular region,

$Δ x = v Δ t$ .

For a triangular region,

$Δ x = \frac{1}{2} \times base \times height$ .

For a speed–time graph, the area gives distance traveled because speed is nonnegative. For a velocity–time graph, adding signed areas gives displacement. Adding the magnitudes of the separate areas gives total distance traveled.

Worked Example: Area Under a Velocity–Time Graph

Imagine a car travels at 6 m/s for 5 s, then at −2 m/s for 3 s. The first rectangular area is $6 \times 5 = 30 m$ . The second signed area is $- 2 \times 3 = - 6 m$ . The displacement is $30 - 6 = 24 m$ . The total distance traveled is $30 + 6 = 36 m$ .

Acceleration–Time Graphs

An acceleration–time graph shows how acceleration changes. Its vertical value tells you acceleration directly. A horizontal line above zero represents constant positive acceleration; a line at zero represents zero acceleration; and a horizontal line below zero represents constant negative acceleration.

The signed area under an acceleration–time graph gives the change in velocity:

$Δ v = a Δ t$

for constant acceleration.

Connecting the Three Main Graphs

Position, velocity, and acceleration graphs are connected by rates of change. The slope of a position–time graph gives velocity. The slope of a velocity–time graph gives acceleration. In the opposite direction, area under a velocity–time graph gives displacement, and area under an acceleration–time graph gives change in velocity.

The image above helps you compare a position–time graph with a velocity–time graph at corresponding moments.

Graph	What the height means	What the slope means	What the signed area means
Position–time	Position	Velocity	No standard Grade 9–10 kinematics quantity
Velocity–time	Velocity	Acceleration	Displacement
Acceleration–time	Acceleration	Rate of change of acceleration, an enrichment idea	Change in velocity

Translating Graph Shapes

If a position–time graph is a straight line with positive slope, the velocity–time graph is a horizontal line above zero. If the position–time graph is horizontal, the velocity–time graph lies on zero. If a velocity–time graph rises linearly, the acceleration–time graph is a horizontal line above zero.

A curved position–time graph that becomes progressively steeper has increasing velocity magnitude. To decide whether that means speeding up, also consider the sign of the slope.

This enrichment image extends the pattern by showing position, velocity, acceleration, and jerk together. At Grades 9–10, focus mainly on position, velocity, and acceleration.

A Reliable Graph-Reading Strategy

Use this sequence whenever you meet a new motion graph:

Axes: Identify the variable and unit on each axis.
Interval: Break the graph into meaningful time intervals.
Slope: Decide whether slope has a physical meaning and calculate it when needed.
Area: Decide whether area has a physical meaning and include signs correctly.
Direction: Distinguish positive motion, negative motion, and rest.
Evidence: Support every statement with a graph feature such as height, slope, or area.

Common Misconceptions

A high point on a position–time graph does not mean high speed. Speed depends on slope, not vertical height.

A horizontal line on a velocity–time graph does not mean the object is stopped. It means velocity is constant. The object is stopped only if the horizontal line lies at zero velocity.

A negative velocity is not automatically a slowing object. Negative velocity indicates direction. An object with negative velocity and negative acceleration can be speeding up because the magnitude of its velocity is increasing.

Negative acceleration does not always mean slowing down. Compare the signs of velocity and acceleration.

Area and slope are different ideas. On a velocity–time graph, slope gives acceleration while area gives displacement.

Measurement and Graph Creation

You can create motion graphs from a simple experiment. Mark a straight path, choose an origin, and record the time at which a walker reaches measured positions. A phone stopwatch, video timestamps, motion sensor, or spreadsheet can help. Plot time horizontally and position vertically. Then use selected intervals to calculate slopes.

To improve data quality, repeat measurements, use consistent units, label axes, include a clear scale, and distinguish measured points from lines that are only guides to the eye. When using video, keep the camera fixed and include an object of known length for scale.

Interactive Tasks

Quiz: Test Your Knowledge

What does the slope of a position–time graph represent? (Velocity) (!Position) (!Acceleration) (!Distance)

What does a horizontal segment on a position–time graph show? (The object is at rest) (!The object has maximum speed) (!The object has constant acceleration) (!The object moves backward)

What does the slope of a velocity–time graph represent? (Acceleration) (!Displacement) (!Distance) (!Position)

What does the signed area under a velocity–time graph represent? (Displacement) (!Acceleration) (!Average speed) (!Position)

What does the area under a speed–time graph represent? (Distance traveled) (!Displacement with direction) (!Acceleration) (!Position)

A velocity–time graph is horizontal at 5 m/s. What is the acceleration? (Zero) (!Five meters per second squared) (!Positive and increasing) (!Negative and decreasing)

An object has negative velocity and negative acceleration. What can happen to its speed? (It can increase) (!It must be zero) (!It must decrease) (!It cannot change)

Which graph feature gives instantaneous velocity on a curved position–time graph? (The tangent slope) (!The area under the graph) (!The vertical intercept) (!The highest point)

What does the signed area under an acceleration–time graph represent? (Change in velocity) (!Total distance) (!Position) (!Average position)

Which statement about a true total-distance–time graph is correct? (It cannot decrease) (!It can have negative total distance) (!Its slope gives acceleration) (!Its area gives velocity)

Memory Game

Position	Location relative to a chosen origin
Velocity	Rate of change of displacement
Acceleration	Rate of change of velocity
Gradient	Another word for slope
Displacement	Directed change from initial to final position
Plateau	Horizontal graph segment showing no change in the vertical quantity

Drag and Drop

Match the correct terms.	Topic
Velocity	Slope of a position–time graph
Acceleration	Slope of a velocity–time graph
Displacement	Signed area under a velocity–time graph
Distance traveled	Area under a speed–time graph
Change in velocity	Signed area under an acceleration–time graph

Match each motion quantity to the graph feature that determines it.

Crossword Puzzle

Position	What quantity tells where an object is relative to an origin?
Velocity	What quantity is the slope of a position–time graph?
Gradient	What is another word for the slope of a graph?
Plateau	What word describes a horizontal section of a graph?
Acceleration	What quantity is the slope of a velocity–time graph?
Displacement	What directed quantity equals signed area under a velocity–time graph?

LearningApps

Cloze Text

Open-Ended Tasks

Easy

Graph Walk: Walk along a straight line at constant speed, stop, and continue; sketch the position–time graph that should describe your movement and explain each segment.
Motion Caption: Choose one motion graph from this course and write a short story that matches its intervals, directions, stops, and changes in speed.
Graph Photo: Create a clearly labeled hand-drawn or digital image of a position–time graph with one positive-slope interval, one horizontal interval, and one negative-slope interval.
Human Graph Interview: Ask a classmate how they would explain slope on a motion graph to a younger student, then improve the explanation using correct units and one example.

Standard

Measured Walk Experiment: Measure position and time for a person walking along a straight path, plot the data, calculate two interval velocities, and discuss measurement uncertainty.
Motion Video Analysis: Record a short safe video of an object moving in one dimension, estimate positions at regular time intervals, and create a position–time graph from your measurements.
Graph Translation Project: Design one piecewise velocity–time graph and produce a matching position–time sketch, explaining how every segment of one graph determines the other.
Transport Observation: Visit or observe a safe public place such as a school corridor, athletics track, station platform, or cycle path and describe one real movement that could be represented by a motion graph without identifying individuals.

Advanced

Multi-Graph Investigation: Create matching position–time, velocity–time, and acceleration–time graphs for a motion with at least four intervals and justify every slope and area relationship.
Data Logger Study: Use a motion sensor, phone sensor, or video-tracking tool to collect motion data, compare measured and calculated quantities, and evaluate sources of error.
Motion Graph Tutorial: Produce a three-minute instructional video that teaches the difference between slope and area on velocity–time graphs using your own example and visuals.
Model Critique: Find a motion graph in a textbook, laboratory sheet, sports analysis, or transport context and evaluate whether its labels, scale, units, and interpretation are scientifically precise.

Learning Assessment

Evidence-Based Interpretation: Given an unfamiliar motion graph with several intervals, write a paragraph that identifies direction, rest, speeding up, and slowing down, citing graph features as evidence.
Slope Calculation: Calculate velocities from two intervals of a position–time graph and compare them using both numerical values and graph steepness.
Area Reasoning: Determine displacement and total distance from a velocity–time graph containing positive and negative regions, then explain why the two results differ.
Graph Translation: Convert a verbal motion description into a velocity–time graph and a matching position–time sketch, then justify the correspondence.
Error Analysis: Diagnose three incorrect statements about a motion graph, correct them, and explain the misconception behind each error.
Transfer Challenge: Use a real or simulated motion dataset to construct a graph, estimate a rate of change, interpret an area where appropriate, and evaluate how measurement uncertainty affects your conclusion.

Evidence of Learning

Evidence type	What successful learning looks like
Knowledge	You distinguish position, distance, displacement, speed, velocity, and acceleration and connect each quantity to the correct graph.
Skills	You read axes and units, calculate slopes and areas, interpret signs, translate between graph types, and explain graph features with precise language.
Products	You create accurate labeled graphs, worked calculations, experimental data displays, written explanations, and optional videos or digital models.
Reasoning	You justify claims using evidence from height, slope, area, interval, sign, and units rather than relying on visual guesses.
Transfer	You apply motion-graph ideas to new experiments, transport examples, sports motion, sensor data, and unfamiliar graph shapes.