Velocity and Acceleration

Introduction

Motion is everywhere: a cyclist speeds up after a traffic light changes, a ball slows as it climbs, a train travels at nearly constant speed, and a satellite changes direction even when its speed stays almost constant. Kinematics is the part of physics that describes motion without first asking what forces cause it. Two of its most important quantities are velocity and acceleration.

This aiMOOC is designed for Grades 9–10. You will learn how to describe motion with words, equations, vectors, tables, and graphs. You will also investigate real motion, interpret evidence, explain sign conventions, and connect mathematical representations to physical situations.

Learning Goals

By the end of the course, you should be able to explain the difference between distance and displacement, distinguish speed from velocity, calculate average velocity and average acceleration, interpret positive and negative signs, read position–time and velocity–time graphs, use constant-acceleration equations in one dimension, and explain why a change of direction counts as acceleration.

You should also be able to plan a simple motion investigation, collect measurements responsibly, represent data clearly, identify uncertainty, and use evidence to justify a conclusion.

Describing Motion

To describe motion, you first choose a reference frame and a positive direction. Imagine a straight hallway. You might define east as positive and west as negative. Positions, displacements, velocities, and accelerations can then carry signs that communicate direction.

Position tells where an object is relative to an origin. Distance is the total path length traveled and is a scalar. Displacement is the change in position and is a vector in one-dimensional motion because its sign indicates direction.

For motion from an initial position $x_{i}$ to a final position $x_{f}$ :

$Δ x = x_{f} - x_{i}$

If you walk 30 m east and then 10 m west, your total distance is 40 m, but your displacement is 20 m east. Distance and displacement therefore answer different questions.

Scalars and Vectors

A scalar has magnitude only. A vector has magnitude and direction. Speed is a scalar; velocity is a vector. Distance is a scalar; displacement is a vector. Acceleration is also a vector.

Quantity	What it describes	Scalar or vector	Common SI unit
Distance	Total path length	Scalar	metre
Displacement	Change in position	Vector	metre
Speed	Rate at which distance changes	Scalar	metre per second
Velocity	Rate at which displacement changes	Vector	metre per second
Acceleration	Rate at which velocity changes	Vector	metre per second squared

A negative vector value does not automatically mean “slower.” It means the vector points opposite to the chosen positive direction. Whether an object is speeding up or slowing down depends on the relationship between the signs or directions of velocity and acceleration.

Velocity

Velocity describes how quickly displacement changes with time and in what direction the motion occurs. Average velocity is

$v_{avg} = \frac{Δ x}{Δ t}$

where $Δ x$ is displacement and $Δ t$ is the elapsed time.

The SI unit of velocity is metres per second, written m/s. A statement such as “8 m/s east” gives both magnitude and direction and therefore describes a velocity. “8 m/s” alone gives a speed unless a direction has already been established by a sign convention.

Average and Instantaneous Velocity

Average velocity uses a displacement measured over a time interval. Instantaneous velocity is the velocity at a particular moment. A car’s speedometer reports the magnitude of instantaneous velocity, not average velocity over an entire journey.

On a position–time graph, the average velocity over an interval equals the slope of the secant line connecting the two endpoints. Instantaneous velocity corresponds to the slope of the tangent at one point.

Fehler beim Erstellen des Vorschaubildes:

Worked Example: Average Velocity

A bicycle moves from 10 m east of an origin to 130 m east of the origin in 20 s.

The displacement is $130 m - 10 m = 120 m$ east.

The average velocity is $120 m / 20 s = 6 m/s$ east.

If the cyclist had taken a winding route but ended at the same place after the same time, the average velocity would still be 6 m/s east, even though the average speed could be larger because the total distance could be larger.

Acceleration

Acceleration describes how quickly velocity changes with time. Because velocity includes direction, an object accelerates when it speeds up, slows down, changes direction, or changes both speed and direction.

Average acceleration is

$a_{avg} = \frac{Δ v}{Δ t} = \frac{v_{f} - v_{i}}{Δ t}$

The SI unit is metres per second squared, written m/s². An acceleration of 3 m/s² in the positive direction means the velocity changes by 3 m/s toward the positive direction during each second, provided the acceleration remains constant.

Datei:Acceleration vectors.svg

Speeding Up and Slowing Down

In one-dimensional motion, comparing the signs of velocity and acceleration is especially useful.

Velocity	Acceleration	What happens to speed
Positive	Positive	Speed increases
Positive	Negative	Speed decreases
Negative	Negative	Speed increases
Negative	Positive	Speed decreases

The rule is simple: when velocity and acceleration point in the same direction, speed increases. When they point in opposite directions, speed decreases. This rule works better than memorizing “positive acceleration means speeding up,” because positive acceleration can occur while an object is slowing down if its velocity is negative.

Worked Example: Braking

Take forward as positive. A car changes velocity from +18 m/s to +6 m/s in 3 s.

$a = \frac{6 - 18}{3} = - 4 {m/s}^{2}$

The negative acceleration points opposite the positive velocity. The car is therefore slowing down. The magnitude of its velocity decreases by 4 m/s each second during this interval.

Motion Graphs

Graphs connect motion to mathematics. The horizontal axis is usually time. The vertical axis may show position, velocity, or acceleration. The same motion looks different on each type of graph, so always identify the axes before interpreting a line.

Position–Time Graphs

On a position–time graph, the slope gives velocity.

A horizontal line means constant position and therefore zero velocity. A straight line with constant positive slope means constant positive velocity. A straight line with constant negative slope means constant negative velocity. A curve whose slope becomes steeper shows changing velocity and therefore nonzero acceleration.

Fehler beim Erstellen des Vorschaubildes:

Velocity–Time Graphs

On a velocity–time graph, the slope gives acceleration. A horizontal line means constant velocity and zero acceleration. An upward slope means positive acceleration; a downward slope means negative acceleration.

The signed area between a velocity–time graph and the time axis gives displacement. Area above the time axis contributes positive displacement, while area below the axis contributes negative displacement.

Fehler beim Erstellen des Vorschaubildes:

Acceleration–Time Graphs

On an acceleration–time graph, a horizontal line above or below zero represents constant nonzero acceleration. A line along zero represents zero acceleration. The signed area under an acceleration–time graph over a time interval gives the change in velocity during that interval.

A motion with constant acceleration therefore has a particularly simple set of related graphs: acceleration is constant, velocity changes linearly with time, and position changes quadratically with time.

Fehler beim Erstellen des Vorschaubildes:

Constant Acceleration in One Dimension

When acceleration is constant, several equations let you connect displacement, velocity, acceleration, and time. These equations should not be applied automatically to motion with changing acceleration.

$v_{f} = v_{i} + a t$

$Δ x = v_{i} t + \frac{1}{2} a t^{2}$

$v_{f}^{2} = v_{i}^{2} + 2 a Δ x$

The symbols mean: $v_{i}$ is initial velocity, $v_{f}$ is final velocity, $a$ is constant acceleration, $t$ is elapsed time, and $Δ x$ is displacement.

Worked Example: Accelerating Scooter

A scooter moves in the positive direction at 2 m/s and accelerates constantly at 1.5 m/s² for 4 s.

Final velocity:

$v_{f} = 2 + (1.5) (4) = 8 m/s$

Displacement:

$Δ x = (2) (4) + \frac{1}{2} (1.5) (4^{2}) = 20 m$

The final velocity is 8 m/s in the positive direction, and the scooter travels a displacement of 20 m during the 4 s interval.

Choosing an Equation

Before substituting numbers, list what you know, identify the unknown, choose a positive direction, include signs, and check whether acceleration is actually constant. After calculating, inspect the unit and ask whether the sign and magnitude make physical sense.

For example, if a vehicle is moving in the positive direction and braking, its acceleration should normally be negative under that sign convention. A result with an unexpected sign can be a useful clue that you made a setup or arithmetic error.

Direction Changes and Circular Motion

Acceleration does not require a change in speed. If an object moves around a curve at constant speed, the direction of its velocity changes continuously. Because velocity changes, the object accelerates.

In uniform circular motion, the instantaneous velocity is tangent to the circular path while the acceleration points inward toward the center. This inward acceleration changes the direction of the velocity vector.

Fehler beim Erstellen des Vorschaubildes:

A turnaround in one dimension also illustrates the idea. Suppose an object moves left, so its velocity is negative, while its acceleration is positive. It first slows down. At the turning point its instantaneous velocity is zero, but its acceleration can still be positive. After the turnaround, its velocity becomes positive and it speeds up.

Free Fall as an Acceleration Example

Near Earth’s surface and when air resistance is neglected, freely falling objects have an approximately constant downward acceleration with magnitude 9.8 m/s². If upward is chosen as positive, the acceleration is approximately −9.8 m/s². If downward is chosen as positive, the same physical acceleration is approximately +9.8 m/s².

The sign depends on your coordinate choice; the physical direction does not. This is why a clearly stated sign convention is essential in kinematics.

A ball thrown upward can have positive upward velocity while its acceleration is downward. It slows on the way up, has zero instantaneous velocity at the highest point, and then speeds up downward. Its acceleration remains downward throughout the idealized flight.

Measuring Motion in the Real World

You can investigate velocity and acceleration with a toy car, a rolling ball, a phone camera, a stopwatch, measuring tape, or motion-sensor software. Good experiments begin with a clear question and a measurable plan.

For a simple video investigation, place a metre stick or another known length in the camera view, keep the camera fixed, record motion from the side, and identify the object’s position at equal time intervals. A position–time table can be converted into a graph. Slopes over intervals estimate average velocity. Changes in those velocity estimates can be used to estimate acceleration.

Real measurements contain uncertainty. Reaction time, frame rate, camera angle, unclear position markers, and uneven surfaces can affect results. Repeating trials and reporting reasonable precision make conclusions more reliable.

Common Misconceptions

Velocity is not the same as speed. Velocity includes direction.

Negative velocity does not mean slow. It means motion in the negative direction chosen for the coordinate system.

Negative acceleration does not always mean slowing down. An object with negative velocity and negative acceleration speeds up.

Zero velocity does not always mean zero acceleration. At the highest point of an ideal vertical throw, velocity is momentarily zero while acceleration is still downward.

Constant speed does not always mean zero acceleration. Motion at constant speed around a curve involves changing velocity direction and therefore acceleration.

A steep position–time graph does not directly show acceleration. Its slope shows velocity; acceleration is related to how that slope changes.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement best describes velocity? (Speed with direction) (!Distance traveled per unit time only) (!A measure of position only) (!A quantity with no direction)

How is average velocity calculated? (Displacement divided by elapsed time) (!Distance multiplied by elapsed time) (!Acceleration divided by displacement) (!Final speed divided by initial speed)

What does acceleration measure? (Change in velocity per unit time) (!Total distance traveled) (!Position relative to an origin) (!Mass per unit volume)

What is the SI unit of acceleration? (Metres per second squared) (!Metres) (!Seconds per metre) (!Metres per second)

What is the acceleration of an object moving with constant velocity in a straight line? (Zero) (!Always positive) (!Always negative) (!Equal to its speed)

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

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

What does the signed area under a velocity time graph represent? (Displacement) (!Acceleration) (!Mass) (!Instantaneous speed only)

When does an object speed up in one dimensional motion? (When velocity and acceleration point in the same direction) (!Whenever acceleration is positive) (!Whenever velocity is negative) (!Whenever acceleration is zero)

Why can an object moving at constant speed in a circle still accelerate? (Its velocity direction changes) (!Its mass changes continuously) (!Its distance traveled becomes zero) (!Its time interval becomes negative)

Memory Game

Velocity	Rate of change of displacement with direction
Acceleration	Rate of change of velocity
Displacement	Change in position from start to finish
Speed	Rate at which distance is traveled
Vector	Quantity with magnitude and direction
Gradient	Steepness used to interpret motion graphs
Kinematics	Study of motion without first considering its causes

Drag and Drop

Match the correct terms.	Topic
Velocity	Rate of change of displacement
Acceleration	Rate of change of velocity
Position time slope	Instantaneous or average velocity
Velocity time slope	Acceleration
Velocity time area	Displacement

...

Crossword Puzzle

Velocity	Which vector quantity describes how fast position changes and in what direction?
Acceleration	Which vector quantity describes how velocity changes with time?
Displacement	What word means the change from initial position to final position?
Gradient	What word describes the steepness of a graph?
Vector	What kind of quantity has both magnitude and direction?
Kinematics	What branch of mechanics describes motion without first considering its causes?

LearningApps

Cloze Text

A change in position is called

. Velocity includes both magnitude and

. Average velocity is displacement divided by elapsed

. Acceleration measures the rate of change of

. On a position–time graph, the

represents velocity. On a velocity–time graph, the slope represents

. Constant velocity in a straight line means the acceleration is

. The signed area under a velocity–time graph gives

. A quantity with magnitude and direction is a

. An object moving at constant speed around a curve still accelerates because its direction

.

Open-Ended Tasks

Easy

Motion Vocabulary Poster: Create a one-page visual poster that explains position, distance, displacement, speed, velocity, and acceleration using your own examples and labeled arrows.
Walking Velocity Trial: Mark a short straight path in a safe area, measure your walking time over a known displacement, calculate average velocity, and explain your sign convention.
Graph Story: Draw a simple position–time graph for a journey with rest, forward motion, and backward motion, then write a short story that matches each segment.
Everyday Acceleration Photo Hunt: Find or create four safe images showing speeding up, slowing down, turning, and constant velocity, then add one sentence explaining the motion in each image.

Standard

Toy Car Investigation: Record a toy car moving along a measured track, estimate positions at equal time intervals, calculate interval velocities, and decide whether the car accelerates.
Motion Graph Video: Produce a two-minute explainer video that compares position–time and velocity–time graphs and demonstrates how slope is interpreted on each.
Interview About Motion: Interview a cyclist, driver, coach, or athlete about situations where changing direction matters as much as changing speed, then connect two interview examples to physics vocabulary.
School Motion Survey: Visit a safe supervised area such as a corridor, gym, or playground, observe three kinds of motion, and classify each using velocity and acceleration with evidence.

Advanced

Video Tracking Project: Use frame-by-frame video data from a safe moving object to build position–time and velocity–time graphs, estimate acceleration, and discuss at least two sources of uncertainty.
Braking Model: Develop a mathematical model for a vehicle that slows with constant acceleration, choose realistic starting values, calculate stopping time and displacement, and explain the limits of your model.
Circular Motion Investigation: Design a demonstration or simulation showing constant speed with changing velocity direction, create a vector diagram, and explain why inward acceleration is required.
Motion Data Critique: Compare two different methods for measuring motion, such as stopwatch timing and video analysis, then write an evidence-based evaluation of accuracy, precision, practicality, and possible bias.

Learning Assessment

Graph Interpretation Assessment: Analyze a multi-segment velocity–time graph, identify intervals of positive, negative, and zero acceleration, calculate displacement from areas, and justify every conclusion.
Representation Transfer: Convert a written description of motion into a position–time graph and a velocity–time graph, then explain how the two graphs communicate the same event differently.
Sign Convention Reasoning: Solve a turnaround problem using a clearly stated positive direction and explain why the signs of velocity and acceleration change or remain constant at each stage.
Experimental Evidence: Use a small set of measured position and time data to estimate velocity and acceleration, identify uncertainty, and decide whether constant acceleration is a reasonable model.
Constant Acceleration Application: Solve a contextual problem using a suitable constant-acceleration equation and defend why that equation is valid for the stated assumptions.
Misconception Challenge: Evaluate the claim “negative acceleration always means slowing down” by producing a counterexample, diagram, and explanation.

Evidence of Learning

Knowledge: You can define and distinguish position, distance, displacement, speed, velocity, acceleration, scalar, vector, and reference frame, and you can state the relationships among motion graphs.

Skills: You can calculate average velocity and acceleration, use signs consistently, interpret slopes and signed areas, apply constant-acceleration equations, make graphs from data, and communicate units correctly.

Products: Strong evidence may include a labeled motion poster, a graph story, a video explanation, a data table, position–time and velocity–time graphs, a vector diagram, a model, or an investigation report.

Reasoning: You can use the directions of velocity and acceleration to decide whether speed increases or decreases, explain turnarounds, and distinguish a zero velocity at one instant from zero acceleration.

Experimental practice: You can plan a fair motion measurement, collect repeatable observations, identify uncertainty, use reasonable precision, and compare evidence with a mathematical model.

Transfer: You can apply velocity and acceleration ideas to unfamiliar situations such as sports, transport, free fall, circular motion, robotics, or amusement rides and justify which representation is most useful.