written by
Gary Gladding
published by
the University of llinois Physics Education Research Group

This interactive homework problem shows a graph of velocity versus time for a moving object and the student is asked to find the final position. The problem is accompanied by a sequence of questions designed to encourage critical thinking and engage students beyond traditional textbook problems. The questions are user-activated, and carefully designed to guide beginning students through conceptual analysis before attempting the mathematics.

This item is part of a larger collection of interactive problems developed by the Illinois Physics Education Research Group.

Editor's Note:Physics education research shows that students often enter college courses with limited understanding of the meaning behind velocity vs. time graphs and position vs. time graphs. This particular tutorial will help students realize that velocity is the time derivative of distance and acceleration is the time derivative of velocity, an important distinction for understanding the physics of motion.

9-12: 2A/H1. Mathematics is the study of quantities and shapes, the patterns and relationships between quantities or shapes, and operations on either quantities or shapes. Some of these relationships involve natural phenomena, while others deal with abstractions not tied to the physical world.

2C. Mathematical Inquiry

9-12: 2C/H3. To be able to use and interpret mathematics well, it is necessary to be concerned with more than the mathematical validity of abstract operations and to take into account how well they correspond to the properties of the things represented.

4. The Physical Setting

4F. Motion

6-8: 4F/M3a. An unbalanced force acting on an object changes its speed or direction of motion, or both.

9. The Mathematical World

9C. Shapes

9-12: 9C/H1. Distances and angles that are inconvenient to measure directly can be found from measurable distances and angles using scale drawings or formulas.

12. Habits of Mind

12B. Computation and Estimation

6-8: 12B/M3. Calculate the circumferences and areas of rectangles, triangles, and circles, and the volumes of rectangular solids.

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

MP.1 Make sense of problems and persevere in solving them.

Geometry (K-8)

Solve real-world and mathematical problems involving area, surface
area, and volume. (6)

6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

Solve real-life and mathematical problems involving angle measure,
area, surface area, and volume. (7)

7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Understand and apply the Pythagorean Theorem. (8)

8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

High School — Functions (9-12)

Interpreting Functions (9-12)

F-IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.^{?}

High School — Geometry (9-12)

Expressing Geometric Properties with Equations (9-12)

G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.^{?}

This resource is part of 3 Physics Front Topical Units.

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

This is a web-based homework problem that helps students understand velocity vs. time graphs (v vs. t). A sequence of user-activated questions guides beginners through a full conceptual analysis before introducing the math. Based on principles of physics education research (PER).

Topic: Kinematics: The Physics of Motion Unit Title: Motion in One Dimension

This is a web-based homework problem that helps students understand velocity vs. time graphs (v vs. t). A sequence of user-activated questions takes beginners through a full conceptual analysis before introducing the math. It was developed using principles of physics education research. Appropriate for gifted/talented middle school students.

Topic: Kinematics: The Physics of Motion Unit Title: Motion in One Dimension

This is a web-based homework problem that helps students understand velocity vs. time graphs (v vs. t). A sequence of user-activated questions guides beginners through a full conceptual analysis before introducing the math. Based on PER principles (physics education research).

<a href="https://www.compadre.org/precollege/items/detail.cfm?ID=6384">Gladding, Gary. Illinois PER Interactive Examples: v versus t. Urbana: University of llinois Physics Education Research Group, June 16, 2006.</a>

G. Gladding, (University of llinois Physics Education Research Group, Urbana, 2006), WWW Document, (https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t).

G. Gladding, Illinois PER Interactive Examples: v versus t, (University of llinois Physics Education Research Group, Urbana, 2006), <https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t>.

Gladding, G. (2006, June 16). Illinois PER Interactive Examples: v versus t. Retrieved January 21, 2021, from University of llinois Physics Education Research Group: https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t

Gladding, Gary. Illinois PER Interactive Examples: v versus t. Urbana: University of llinois Physics Education Research Group, June 16, 2006. https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t (accessed 21 January 2021).

Gladding, Gary. Illinois PER Interactive Examples: v versus t. Urbana: University of llinois Physics Education Research Group, 2006. 16 June 2006. 21 Jan. 2021 <https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t>.

@misc{
Author = "Gary Gladding",
Title = {Illinois PER Interactive Examples: v versus t},
Publisher = {University of llinois Physics Education Research Group},
Volume = {2021},
Number = {21 January 2021},
Month = {June 16, 2006},
Year = {2006}
}

%A Gary Gladding %T Illinois PER Interactive Examples: v versus t %D June 16, 2006 %I University of llinois Physics Education Research Group %C Urbana %U https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t %O text/html

%0 Electronic Source %A Gladding, Gary %D June 16, 2006 %T Illinois PER Interactive Examples: v versus t %I University of llinois Physics Education Research Group %V 2021 %N 21 January 2021 %8 June 16, 2006 %9 text/html %U https://per.physics.illinois.edu/per/IE/ie.pl?phys111/ie/01/IE_v_vs_t

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A link to the full collection of interactive homework tutorials by the same author. Included are problems designed for a variety of physics courses, including preparatory, algebra-based, and calculus-based. Topics include mechanics, light, electricity and magnetism, thermal physics, waves and quantum mechanics, and modern physics.