Motion in One Dimension

Introduction

Motion in one dimension is the study of motion along a single straight line. A runner moving along a straight track, an elevator moving vertically, and a train traveling along a straight section of rail can all be modeled with one coordinate. In physics, this description of motion is part of kinematics. You focus on where an object is, how fast and in which direction it moves, and how its velocity changes.

This aiMOOC is designed for Grades 9–10. You will use words, measurements, diagrams, graphs, and algebra to connect position, displacement, distance, speed, velocity, acceleration, and free fall.

Learning Goals

By the end of the course, you should be able to distinguish scalars from vectors, choose a positive direction, calculate displacement, average velocity, average speed, and average acceleration, interpret motion graphs, and solve basic constant-acceleration and free-fall problems. You should also be able to explain what the sign of a quantity means instead of treating a negative sign as automatically meaning “slowing down.”

Describing Position and Displacement

To describe one-dimensional motion, choose an origin and a positive direction. Positions to one side of the origin are positive and positions to the other side are negative. The coordinate is often called x for horizontal motion and y for vertical motion.

Position tells you where an object is relative to the origin. If a cart is at x = +6 m, it is six metres in the chosen positive direction from the origin.

Displacement is the change in position:

Δx = x_final − x_initial

Displacement can be positive, negative, or zero. It depends only on the starting and ending positions, not on the path traveled. Distance traveled is different: it is the total length of the path and is never negative.

Example: You walk from x = 2 m to x = 14 m. Your displacement is +12 m. If you then walk back to x = 5 m, your displacement for the whole trip is +3 m, even though the total distance traveled is 21 m.

Scalars and Vectors in One Dimension

A scalar has magnitude only. Distance, speed, and time are scalars. A vector has magnitude and direction. Displacement, velocity, and acceleration are vectors. In one dimension, direction can be represented efficiently by a plus or minus sign after you declare which direction is positive.

A sign convention is a choice, not a law of nature. If you choose right as positive, left is negative. If you choose upward as positive, downward is negative. Keep the same convention throughout one problem.

Speed and Velocity

Average speed is total distance divided by elapsed time:

average speed = total distance / elapsed time

Average velocity is displacement divided by elapsed time:

v_avg = Δx / Δt

Velocity contains direction; speed does not. If you travel 100 m east and then 100 m west in 50 s, the distance is 200 m, so your average speed is 4 m/s. Your displacement is zero, so your average velocity is 0 m/s.

Instantaneous speed and instantaneous velocity describe motion at a particular moment. A car speedometer approximates instantaneous speed. In one dimension, the magnitude of instantaneous velocity is instantaneous speed.

Acceleration

Acceleration describes how quickly velocity changes:

a_avg = Δv / Δt

The SI unit of acceleration is metres per second squared, written m/s². Positive acceleration points in the chosen positive direction; negative acceleration points in the chosen negative direction.

Do not confuse the sign of acceleration with speeding up or slowing down. An object speeds up when velocity and acceleration have the same sign. It slows down when they have opposite signs. For example, an object moving left can have negative velocity and positive acceleration; in that case, it slows down.

A Numerical Acceleration Example

A cart changes velocity from +4 m/s to +10 m/s in 3 s. Its average acceleration is:

a = (10 m/s − 4 m/s) / 3 s = +2 m/s²

The positive sign means the acceleration points in the positive direction. Because the velocity is also positive, the cart is speeding up.

Motion Graphs

Graphs let you see how motion changes with time. The horizontal axis is usually time. The vertical axis identifies the motion quantity being graphed.

Position-Time Graphs

On a position-time graph, the slope tells you velocity. A horizontal line has zero slope, so the object is at rest. A straight line with positive slope represents constant positive velocity, while a straight line with negative slope represents constant negative velocity. A curve with changing slope represents changing velocity.

A steeper position-time graph means a greater speed because the magnitude of the slope is larger. The graph’s height tells you position; it does not directly tell you speed.

Velocity-Time Graphs

On a velocity-time graph, the slope tells you acceleration. The signed area between the velocity curve and the time axis gives displacement over that time interval.

A horizontal line above zero means constant positive velocity and zero acceleration. A line sloping upward means positive acceleration, while a line sloping downward means negative acceleration. A graph below the time axis represents negative velocity.

Connecting Position, Velocity, and Acceleration Graphs

The same motion can be represented by three different graphs. For constant velocity, position changes linearly, velocity is constant, and acceleration is zero. For constant nonzero acceleration, velocity changes linearly with time and the position-time graph curves.

When reading motion graphs, always check the axis labels and units first. Then ask what the slope means, whether the graph crosses zero, and whether the graph is above or below the time axis.

Constant Acceleration Equations

When acceleration is constant, a small set of equations connects displacement, velocity, acceleration, and elapsed time. Use one consistent sign convention.

v_f = v_i + at

Δx = v_it + 1/2 at²

v_f² = v_i² + 2aΔx

Δx = 1/2(v_i + v_f)t

These equations are valid for constant acceleration. They are not automatically valid when acceleration changes during the interval.

A Problem-Solving Routine

Coordinate system: Choose the positive direction and write it down.
Physical quantity: List known quantities with signs and units, then identify the unknown.
Kinematic equation: Select an equation that contains the known quantities and the unknown.
Algebra: Rearrange before inserting numbers when that makes the reasoning clearer.
Dimensional analysis: Check units, sign, and whether the size of the answer is physically reasonable.

Worked Example: Speeding Up

A cyclist has initial velocity +4 m/s and constant acceleration +2 m/s² for 3 s.

Final velocity:

v_f = 4 m/s + (2 m/s²)(3 s) = 10 m/s

Displacement:

Δx = (4 m/s)(3 s) + 1/2(2 m/s²)(3 s)² = 21 m

The cyclist moves 21 m in the positive direction and finishes at 10 m/s.

Worked Example: Braking

A car moving at +18 m/s brakes with constant acceleration −3 m/s² until it stops.

Using v_f = v_i + at:

0 = 18 m/s + (−3 m/s²)t

t = 6 s

Its displacement while braking is:

Δx = 1/2(18 m/s + 0 m/s)(6 s) = 54 m

The negative acceleration does not mean the car moves backward. Here the velocity remains positive until the car stops; the acceleration points opposite the motion and reduces the speed.

Free Fall as One-Dimensional Motion

Free fall is motion in which gravity is the only significant influence on the object’s motion. Near Earth’s surface, and when air resistance can be neglected, the acceleration has nearly constant magnitude:

g ≈ 9.81 m/s²

If upward is positive, the acceleration is a = −g. If downward is positive, the acceleration is a = +g. The physics is the same; only the sign convention changes.

The increasing spacing between successive images of the falling ball shows that the ball covers more distance in equal time intervals: its speed is increasing.

A ball thrown upward is already in free fall after it leaves the hand, if air resistance is neglected. At the highest point its velocity is momentarily zero, but its acceleration is still downward with magnitude g.

Worked Example: Dropped Object

A small object is dropped from rest and falls for 2.0 s. Choose upward as positive, so v_i = 0 and a = −9.8 m/s².

Velocity after 2.0 s:

v_f = 0 + (−9.8 m/s²)(2.0 s) = −19.6 m/s

Displacement after 2.0 s:

Δy = 0 + 1/2(−9.8 m/s²)(2.0 s)² = −19.6 m

The negative signs mean the velocity and displacement are downward relative to the chosen positive direction.

Galileo and the Idea of Equal Free-Fall Acceleration

A famous reasoning strategy associated with Galileo Galilei challenges the claim that heavier objects must fall faster simply because they are heavier. Modern physics shows that, in the absence of air resistance, objects at the same location have the same gravitational acceleration regardless of mass. Differences seen in ordinary air, such as between a feather and a stone, can result from air resistance.

Measurement, Models, and Uncertainty

Real measurements are never perfectly exact. Stopwatch reaction time, video frame rate, ruler precision, camera angle, and inconsistent release technique can affect data. A useful physics model makes assumptions explicit. In this course, common assumptions include motion along one line, negligible air resistance, and constant acceleration over the interval being studied.

When you compare a model with data, do not only ask whether the numbers match exactly. Ask whether the pattern is consistent with the model within reasonable measurement uncertainty.

Interactive Tasks

Quiz: Test Your Knowledge

Which quantity is final position minus initial position? (Displacement) (!Distance) (!Speed) (!Time)

If right is positive, what is the displacement from x = 2 m to x = 14 m? (Positive twelve meters) (!Negative twelve meters) (!Positive sixteen meters) (!Zero meters)

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

A velocity changes from 4 m/s to 10 m/s in 3 s. What is the average acceleration? (Two meters per second squared) (!Three meters per second squared) (!Six meters per second squared) (!Fourteen meters per second squared)

What is the acceleration of an object moving with constant velocity? (Zero) (!Constant positive) (!Constant negative) (!Increasing)

What does the signed area under a velocity-time graph represent? (Displacement) (!Acceleration) (!Position) (!Speed)

A ball thrown upward reaches its highest point. Ignoring air resistance, what is true at that instant? (Velocity is zero while acceleration is downward) (!Velocity and acceleration are both zero) (!Velocity is upward while acceleration is zero) (!Velocity is downward while acceleration is upward)

What is the approximate magnitude of free-fall acceleration near Earth’s surface? (Nine point eight one meters per second squared) (!One meter per second squared) (!Three meters per second squared) (!Ninety eight meters per second squared)

What distinguishes velocity from speed? (Velocity includes direction) (!Velocity has no units) (!Speed includes direction) (!Speed can be negative)

A cart starts at 5 m/s and accelerates at 2 m/s squared for 3 s. What is its final velocity? (Eleven meters per second) (!Seven meters per second) (!Ten meters per second) (!Fifteen meters per second)

Memory Game

Position	Location relative to a chosen origin
Displacement	Signed change from initial to final position
Speed	Distance traveled per elapsed time
Velocity	Displacement per elapsed time with direction
Acceleration	Rate at which velocity changes
Free fall	Motion governed only by gravity in the ideal model
Slope	Rise divided by run on a graph

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
Rest	Horizontal line on a position-time graph
Constant velocity	Horizontal line on a velocity-time graph

Match each motion idea with the graph feature that represents it.

Crossword Puzzle

Position	What word describes an object’s location relative to an origin?
Velocity	Which quantity combines rate of position change with direction?
Acceleration	What word names the rate of change of velocity?
Displacement	What is the signed change from initial to final position called?
Kinematics	What branch of mechanics describes motion without first focusing on its causes?
Gravity	What interaction causes free-fall acceleration near Earth?

LearningApps

Cloze Text

One-dimensional motion can be described with a single

. The signed change from an initial position to a final position is

. Average velocity equals displacement divided by

. Average speed uses total

instead of displacement. Acceleration measures the rate of change of

. On a position-time graph, velocity is represented by the

. On a velocity-time graph, the signed area under the curve gives

. Constant-acceleration equations apply only when acceleration is treated as

. Near Earth’s surface, ideal free fall has an acceleration magnitude of about

. At the highest point of a ball thrown upward, its velocity is zero but its acceleration remains directed

.

Open-Ended Tasks

Easy

Motion diary: Observe one safe example of straight-line motion, such as an elevator or a person walking down a hallway. Record the origin, positive direction, start position, end position, elapsed time, and a short explanation of the motion.
Position-time graph: Walk along a measured straight path while a partner records your position every two seconds. Draw a position-time graph and describe what each segment means.
Motion storyboard: Create a six-frame drawing or photo sequence of an object moving along one line. Add arrows and signed coordinates to show position and direction without revealing anyone’s private information.
Graph narration: Choose a simple position-time graph and record a one-minute audio or video explanation of when the object is at rest, moves positively, or moves negatively.

Standard

Toy car investigation: Use a toy car on a straight, level path. Measure position at equal time intervals, estimate average velocities, graph the data, and discuss measurement uncertainty.
Video motion analysis: Record a safe straight-line motion with a visible scale, use the frame rate to estimate positions and times, and compare a position-time graph with a velocity-time graph.
Free-fall experiment: With teacher supervision, drop a small soft object from a modest height and use slow-motion video to estimate how the spacing changes between equal time intervals. Explain why the result is consistent or inconsistent with acceleration.
Physics interview: Interview a driver, cyclist, coach, engineer, or technician about how they judge speed, stopping, or timing. Compare everyday language with the physics meanings of speed, velocity, and acceleration.

Advanced

Braking model: Build a mathematical model for a bicycle or car slowing with constant acceleration. Predict stopping time and displacement for at least three initial velocities, then explain what the model leaves out.
Motion sensor design: Design a classroom procedure using a phone sensor, video tool, or motion detector to collect one-dimensional motion data. Specify variables, units, calibration, expected graphs, and sources of uncertainty.
Model comparison: Collect or use a teacher-provided motion dataset and compare a constant-velocity model with a constant-acceleration model. Decide which model fits better and justify your decision using graphs and residual differences.
Public motion explainer: Produce a two- to three-minute educational video or illustrated article that corrects one common misconception, such as “negative acceleration always means slowing down” or “acceleration is zero at the top of a throw.” Support the explanation with a graph, equation, and example.

Learning Assessment

Graph-to-story assessment: Given an unfamiliar position-time graph with several segments, write a coherent motion story and justify each part using the sign and magnitude of the slope.
Story-to-graph assessment: Turn a written journey along a straight line into position-time and velocity-time graphs, then explain where your graphs agree with the description.
Equation-choice assessment: For three constant-acceleration situations, identify known and unknown quantities, choose an appropriate kinematic equation, and explain why the unused equations are less efficient.
Free-fall transfer assessment: Analyze a ball thrown vertically upward using a declared sign convention, and explain why zero velocity at the highest point does not mean zero acceleration.
Experimental reasoning assessment: Evaluate a set of position measurements from a moving cart, decide whether constant velocity or constant acceleration is the better model, and support your conclusion with calculations and graph evidence.
Safety and realism assessment: Compare a simple constant-deceleration stopping model with real road braking, identifying at least three factors that could make real stopping distance differ from the model.

Evidence of Learning

Area	Evidence you can provide
Knowledge	You distinguish position, distance, displacement, speed, velocity, acceleration, and free fall; you explain sign conventions and the conditions for constant-acceleration equations.
Skills	You measure motion, calculate rates, use units consistently, construct and interpret motion graphs, select equations, and evaluate whether an answer is reasonable.
Products	You produce clear graphs, a short investigation report, a worked solution set, and at least one visual, audio, or video explanation of motion.
Transfer	You apply one-dimensional motion ideas to unfamiliar contexts such as elevators, sports, vehicle braking, laboratory carts, and falling objects while stating the limits of the model.

OERs on the Topic

The English Wikipedia article on Linear motion provides a compact overview of motion along a straight line.

For further open learning, you can also use OpenStax College Physics: One-Dimensional Kinematics and OpenStax University Physics: Free Fall.

Linked Learning Areas

Motion in One Dimension

This topic connects physics with algebra, graphing, measurement, data analysis, scientific modeling, transportation, and sports science. These links help you transfer straight-line motion ideas from classroom problems to laboratory investigations and everyday situations.

aiMOOC Projects

Diesen aiMOOC als Battle spielen: Mathematik, Klasse 9-10, G-Niveau

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