This page contains a set of 21 problems (with solutions) on motion and forces, designed to support an introductory physics curriculum. Topics encompass kinematics, vector algebra, momentum changes, and more. Each problem can be viewed separately, with detailed solutions, or downloaded as a pdf file. The site includes videos showing solutions to many of the problems, which can be followed with RSS or as podcasts from iTunes. Teachers are encouraged to submit their own problems and solutions to the web site.

Editor's Note: This material is organized to support the Matter and Interactions textbook by Ruth Chabay and Bruce Sherwood, but most of the problems can be used with a standard introductory physics curriculum. Additional information about the pedagogy behind Matter and Interactions, which emphasizes student modeling and a modern perspective on physics, can be found at the Matter and Interactions Home Page

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Aaron Titus

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December 4, 2009
by Alea Smith

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.

4. The Physical Setting

4F. Motion

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/H4. Whenever one thing exerts a force on another, an equal amount of force is exerted back on it.

9-12: 4F/H7. In most familiar situations, frictional forces complicate the description of motion, although the basic principles still apply.

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

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

9-12: 9B/H5. When a relationship is represented in symbols, numbers can be substituted for all but one of the symbols and the possible value of the remaining symbol computed. Sometimes the relationship may be satisfied by one value, sometimes by more than one, and sometimes not at all.

12. Habits of Mind

12B. Computation and Estimation

9-12: 12B/H2. Find answers to real-world problems by substituting numerical values in simple algebraic formulas and check the answer by reviewing the steps of the calculation and by judging whether the answer is reasonable.

Next Generation Science Standards

Motion and Stability: Forces and Interactions (HS-PS2)

Students who demonstrate understanding can: (9-12)

Analyze data to support the claim that Newton's second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. (HS-PS2-1)

Use mathematical representations to support the claim that the total momentum of a system of objects is conserved when there is no net force on the system. (HS-PS2-2)

Disciplinary Core Ideas (K-12)

Forces and Motion (PS2.A)

Newton's second law accurately predicts changes in the motion of macroscopic objects. (9-12)

Momentum is defined for a particular frame of reference; it is the mass times the velocity of the object. (9-12)

If a system interacts with objects outside itself, the total momentum of the system can change; however, any such change is balanced by changes in the momentum of objects outside the system. (9-12)

Crosscutting Concepts (K-12)

Scale, Proportion, and Quantity (3-12)

Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another (e.g., linear growth vs. exponential growth). (9-12)

Stability and Change (2-12)

Change and rates of change can be quantified and modeled over very short or very long periods of time. Some system changes are irreversible. (9-12)

NGSS Science and Engineering Practices (K-12)

Using Mathematics and Computational Thinking (5-12)

Mathematical and computational thinking at the 9â12 level builds on Kâ8 and progresses to using algebraic thinking and analysis, a range of linear and nonlinear functions including trigonometric functions, exponentials and logarithms, and computational tools for statistical analysis to analyze, represent, and model data. Simple computational simulations are created and used based on mathematical models of basic assumptions. (9-12)

Use mathematical representations of phenomena to describe explanations. (9-12)

Common Core State Standards for Mathematics Alignments

Standards for Mathematical Practice (K-12)

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

MP.2 Reason abstractly and quantitatively.

High School â Algebra (9-12)

Reasoning with Equations and Inequalities (9-12)

A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A-REI.4.b Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

High School â 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.^{?}

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

F-IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-IF.7.a Graph linear and quadratic functions and show intercepts, maxima, and minima.

F-IF.8.b Use the properties of exponents to interpret expressions for exponential functions.

F-IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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).

F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

Trigonometric Functions (9-12)

F-TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{?}

This resource is part of a Physics Front Topical Unit.

Topic: Dynamics: Forces and Motion Unit Title: Applications of Newton's Laws

Robust set of 21 problems (with detailed solutions) to give students deep practice in motion, force interactions, and momentum. You can view each problem separately or download as a pdf file. The questions were carefully crafted to align with best practices in physics education. The resource follows the pedagogy of the Matter and Interactions instructional method, which emphasizes student modeling and integration of modern physics into a standard curriculum.

<a href="http://www.compadre.org/precollege/items/detail.cfm?ID=9691">Titus, Aaron. Matter & Interactions Practice Problems: Interactions and Motion. July 11, 2009.</a>

Titus, A. (2009, July 11). Matter & Interactions Practice Problems: Interactions and Motion. Retrieved March 2, 2015, from http://linus.highpoint.edu/~atitus/mandi/index.php?dcsid=1100

Titus, Aaron. Matter & Interactions Practice Problems: Interactions and Motion. July 11, 2009. http://linus.highpoint.edu/~atitus/mandi/index.php?dcsid=1100 (accessed 2 March 2015).

Titus, Aaron. Matter & Interactions Practice Problems: Interactions and Motion. 2009. 11 July 2009. 2 Mar. 2015 <http://linus.highpoint.edu/~atitus/mandi/index.php?dcsid=1100>.

%A Aaron Titus %T Matter & Interactions Practice Problems: Interactions and Motion %D July 11, 2009 %U http://linus.highpoint.edu/~atitus/mandi/index.php?dcsid=1100 %O text/html

%0 Electronic Source %A Titus, Aaron %D July 11, 2009 %T Matter & Interactions Practice Problems: Interactions and Motion %V 2015 %N 2 March 2015 %8 July 11, 2009 %9 text/html %U http://linus.highpoint.edu/~atitus/mandi/index.php?dcsid=1100

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American Journal of Physics article which discusses the pedagogy behind the Matter and Interactions curriculum, authored by Ruth Chabay and Bruce Sherwood. May be freely downloaded in pdf format.