This tutorial, developed for high school physics students, uses multiple graphs and animations to study the relationship between the motion of an object and its graph of Velocity vs. Time. Users explore the relationship between position and velocity, positive and negative velocities, slope and shape of graphs, and acceleration. Interactive self-evaluations are included. See Related Materials for an accompanying lab by the same author.

This item is part of The Physics Classroom, a comprehensive set of tutorials and multimedia resources for high school physics.

Editor's Note:Education research indicates that many students have difficulty differentiating velocity and acceleration, and often plot velocity graphs as the path of an object. See Related Materials for a free research-based diagnostic tool to probe misconceptions related to velocity.

6-8: 4F/M3b. If a force acts towards a single center, the object's path may curve into an orbit around the center.

9-12: 4F/H1. The change in motion (direction or speed) of an object is proportional to the applied force and inversely proportional to the mass.

9-12: 4F/H8. Any object maintains a constant speed and direction of motion unless an unbalanced outside force acts on it.

9. The Mathematical World

9B. Symbolic Relationships

6-8: 9B/M3. Graphs can show a variety of possible relationships between two variables. As one variable increases uniformly, the other may do one of the following: increase or decrease steadily, increase or decrease faster and faster, get closer and closer to some limiting value, reach some intermediate maximum or minimum, alternately increase and decrease, increase or decrease in steps, or do something different from any of these.

9-12: 9B/H4. Tables, graphs, and symbols are alternative ways of representing data and relationships that can be translated from one to another.

9C. Shapes

9-12: 9C/H3c. A graph represents all the values that satisfy an equation, and if two equations have to be satisfied at the same time, the values that satisfy them both will be found where the graphs intersect.

Common Core State Standards for Mathematics Alignments

Expressions and Equations (6-8)

Represent and analyze quantitative relationships between
dependent and independent variables. (6)

6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

Understand the connections between proportional relationships,
lines, and linear equations. (8)

8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Functions (8)

Use functions to model relationships between quantities. (8)

8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.^{?}

Linear, Quadratic, and Exponential Models^{?} (9-12)

F-LE.1.b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

F-LE.1.c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

F-LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Common Core State Reading Standards for Literacy in Science and Technical Subjects 6—12

Craft and Structure (6-12)

RST.9-10.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9—10 texts and topics.

Range of Reading and Level of Text Complexity (6-12)

RST.9-10.10 By the end of grade 10, read and comprehend science/technical texts in the grades 9—10 text complexity band independently and proficiently.

This resource is part of a Physics Front Topical Unit.

Topic: Kinematics: The Physics of Motion Unit Title: Graphing

A companion to the resource above, this online tutorial explores the importance of the slope of v-t graphs as a representation of an object's acceleration. Self-guided evaluations help students overcome common misconceptions.

<a href="http://www.thephysicsfront.org/items/detail.cfm?ID=3313">Henderson, Tom. The Physics Classroom: The Meaning of Shape for a v-t Graph. June 1, 2011.</a>

Henderson, T. (2011, June 1). The Physics Classroom: The Meaning of Shape for a v-t Graph. Retrieved December 4, 2016, from http://www.physicsclassroom.com/Class/1DKin/U1L4a.cfm

Henderson, Tom. The Physics Classroom: The Meaning of Shape for a v-t Graph. June 1, 2011. http://www.physicsclassroom.com/Class/1DKin/U1L4a.cfm (accessed 4 December 2016).

Henderson, Tom. The Physics Classroom: The Meaning of Shape for a v-t Graph. 2004. 1 June 2011. 4 Dec. 2016 <http://www.physicsclassroom.com/Class/1DKin/U1L4a.cfm>.

@misc{
Author = "Tom Henderson",
Title = {The Physics Classroom: The Meaning of Shape for a v-t Graph},
Volume = {2016},
Number = {4 December 2016},
Month = {June 1, 2011},
Year = {2004}
}

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An annotated list of documented student misconceptions related to concepts of position, velocity, and acceleration. Contains probative questions to elicit and address the misconceptions.