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{{T}}
[[Category:English]]
[[Category:Motion in One Dimension]]

{{BR}}
= 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 [[English:Kinematics|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.

[[File:1-D kinematics.svg|500px|frameless|center]]

{{#ev:youtube|https://www.youtube.com/watch?v=ZM8ECpBuQYE|500|center}}

{{BR}}
== Learning Goals ==

By the end of the course, you should be able to distinguish [[English:Scalar (physics)|scalars]] from [[English:Euclidean vector|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.”

{{BR}}
= 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 = xfinal − xinitial

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.

[[File:Vector-as-displacement.svg|500px|frameless|center]]

'''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.

{{BR}}
== 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.

{{BR}}
= 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:

vavg = 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.

[[File:Mean-velocity-in-a-position-time-graph.svg|500px|frameless|center]]

{{BR}}
= Acceleration =

'''Acceleration''' describes how quickly velocity changes:

aavg = 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.

{{#ev:youtube|https://www.youtube.com/watch?v=FOkQszg1-j8|500|center}}

{{BR}}
== 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.

{{BR}}
= 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.

{{BR}}
== 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.

[[File:Example-position-time-diagramm.svg|500px|frameless|center]]

{{#ev:youtube|https://www.youtube.com/watch?v=Mjnu5ePzXDM|500|center}}

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.

{{BR}}
== 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.

[[File:Example-velocity-time-diagramm.svg|500px|frameless|center]]

[[File:Velocity vs time graph.svg|500px|frameless|center]]

{{BR}}
== 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.

[[File:Uniform-motion.svg|500px|frameless|center]]

[[File:Uniform-acceleration.svg|500px|frameless|center]]

{{#ev:youtube|https://www.youtube.com/watch?v=fhOqbAF1Uis|500|center}}

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.

{{BR}}
= Constant Acceleration Equations =

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

vf = vi + at

Δx = vit + 1/2 at²

vf² = vi² + 2aΔx

Δx = 1/2(vi + vf)t

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

{{BR}}
== A Problem-Solving Routine ==

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

{{BR}}
== Worked Example: Speeding Up ==

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

Final velocity:

vf = 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.

{{BR}}
== Worked Example: Braking ==

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

Using vf = vi + 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.

{{BR}}
= 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.

[[File:Falling ball.jpg|500px|frameless|center]]

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.

{{#ev:youtube|https://www.youtube.com/watch?v=vyvDzI22sOE|500|center}}

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.

{{BR}}
== Worked Example: Dropped Object ==

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

Velocity after 2.0 s:

vf = 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.

{{BR}}
== Galileo and the Idea of Equal Free-Fall Acceleration ==

A famous reasoning strategy associated with [[English:Galileo Galilei|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.

[[File:Thought-experiment-free-falling-bodies.svg|500px|frameless|center]]

{{#ev:youtube|https://www.youtube.com/watch?v=GDUdUumkv0o|500|center}}

{{BR}}
= 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.

{{BR}}
= Interactive Tasks =

{{BR}}
== Quiz: Test Your Knowledge ==

{{MC}}

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

{{E}}
 

{{MC}}

'''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)

{{E}}
 

{{MC}}

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

{{E}}
 

{{MC}}

'''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)

{{E}}
 

{{MC}}

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

{{E}}
 

{{MC}}

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

{{E}}
 

{{MC}}

'''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)

{{E}}
 

{{MC}}

'''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)

{{E}}
 

{{MC}}

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

{{E}}
 

{{MC}}

'''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)

{{E}}
 

{{BR}}
== Memory Game ==

<div class="memo-quiz">
{|
|-
| 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
|}
{{E}}
 

{{BR}}
== Drag and Drop ==

<div class="lueckentext-quiz">

{| class="wikitable"
! 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
|}
{{E}}

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

{{BR}}
== Crossword Puzzle ==

<div class="kreuzwort-quiz">
{|
|-
| 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?
|}
{{E}}
 

{{BR}}
== LearningApps ==

<iframe> https://learningapps.org/index.php?s=Motion+in+One+Dimension </iframe>

{{BR}}
== Cloze Text ==

<quiz display=simple>
{'''Complete the text.''' 
|type="{}"}
One-dimensional motion can be described with a single { coordinate }. The signed change from an initial position to a final position is { displacement }. Average velocity equals displacement divided by { elapsed time }. Average speed uses total { distance } instead of displacement. Acceleration measures the rate of change of { velocity }. On a position-time graph, velocity is represented by the { slope }. On a velocity-time graph, the signed area under the curve gives { displacement }. Constant-acceleration equations apply only when acceleration is treated as { constant }. Near Earth’s surface, ideal free fall has an acceleration magnitude of about { 9.81 meters per second squared }. At the highest point of a ball thrown upward, its velocity is zero but its acceleration remains directed { downward }.
</quiz>

 

{{BR}}
= Open-Ended Tasks =

{{BR}}
=== Easy ===
# [[English:Motion diary|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.
# [[English:Position-time graph|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.
# [[English:Motion storyboard|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.
# [[English:Graph narration|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.

{{BR}}
=== Standard ===
# [[English:Toy car investigation|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.
# [[English:Video motion analysis|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.
# [[English:Free-fall experiment|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.
# [[English:Physics interview|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.

{{BR}}
=== Advanced ===
# [[English:Braking model|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.
# [[English:Motion sensor design|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.
# [[English:Model comparison|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.
# [[English:Public motion explainer|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.

{{:Open Task - Create a MOOC}}

{{BR}}
= Learning Assessment =

# [[English:Graph-to-story 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.
# [[English:Story-to-graph assessment|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.
# [[English:Equation-choice assessment|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.
# [[English:Free-fall transfer assessment|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.
# [[English:Experimental reasoning assessment|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.
# [[English:Safety and realism assessment|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.

 
 

{{BR}}
= Evidence of Learning =

{| class="wikitable"
! 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.
|}

 
 

{{BR}}
= OERs on the Topic =

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

<iframe> https://en.m.wikipedia.org/wiki/Linear_motion </iframe>

For further open learning, you can also use [https://openstax.org/books/college-physics-2e/pages/2-introduction-to-one-dimensional-kinematics OpenStax College Physics: One-Dimensional Kinematics] and [https://openstax.org/books/university-physics-volume-1/pages/3-5-free-fall OpenStax University Physics: Free Fall].

 

{{BR}}
= Linked Learning Areas =

{| align=center
{{:D-Tab}}
'''[[English:Motion in One Dimension|Motion in One Dimension]]'''
# [[English:Kinematics|Kinematics]]
# [[English:Position|Position]]
# [[English:Displacement|Displacement]]
# [[English:Speed|Speed]]
# [[English:Velocity|Velocity]]
# [[English:Acceleration|Acceleration]]
# [[English:Motion graph|Motion graph]]
# [[English:Equations of motion|Equations of motion]]
# [[English:Free fall|Free fall]]
# [[English:Measurement uncertainty|Measurement uncertainty]]
|}

This topic connects [[English:Physics|physics]] with [[English:Algebra|algebra]], [[English:Graph of a function|graphing]], [[English:Measurement|measurement]], [[English:Data analysis|data analysis]], [[English:Scientific modelling|scientific modeling]], [[English:Transport|transportation]], and [[English:Sports science|sports science]]. These links help you transfer straight-line motion ideas from classroom problems to laboratory investigations and everyday situations.

{{BR}}
= aiMOOC Projects =
[[Category:English]]
[[Category:Motion in One Dimension]]
[[Category:Physics]]
[[Category:Kinematics]]
[[Category:Mechanics]]
[[Category:Science]]
[[Category:Mathematics]]
[[Category:Grades 9-10]]
[[Category:Secondary education]]
[[Category:AI_MOOC]]
[[Category:GPT aiMOOC]]
{{MT}}

English:Motion in One Dimension - Versionsgeschichte

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