Main Body
SECTION 1
Reading and Solving Word Problems in 1st-year Physics
This introductory unit begins by providing strategies for interpreting and solving physics problems. In the second part of this unit, physics problems are used to learn about how language organizes physics knowledge and practices into units and scales of meaning-making. Both these sections of Unit 1 are foundational for the textbook. As noted above in “How to use this textbook”, this textbook is task-based: after doing the tasks, check the feedback on your work to build on what you know and can do in physics.
Task 1.1: Reading Physics Word Problems
Physics problems come in many shapes and sizes; however, they all typically include a mix of language, figures, symbols. Developing strategies to read, organize and interpret a problem will set you up to successfully solve any physics word problem!
In the module below, read through the Cold Run problem, noting what you do and think about in preparing to solve the problem; that is, pay attention to the strategies you use in interpreting the problem in preparation for solving it. List these strategies you’ve used. We will come back to these strategies later in this chapter.
Task 1.2: Interpreting Word Problems
Now that you have noted the strategies you’ve used for interpreting the problem, proceed to the next task to review six recommended strategies. Reflect on the similarities and differences between these recommended strategies and the strategies you noted in task 1.1.1.
Task 1.3: Solving the Problem & Writing a Solution with Rationale
With the information gathered in task 1.1.1 and 1.1.2 attempt to solve The Cold Run problem.
The 5-Stage Strategy for Solving Physics Problems
The first step or stage in approaching a word problem is to read it through, identifying and extracting key information from the problem. This stage begins the well-known 5-stage strategy to solving physics word problems.
Task 1.4: Thinking Ahead about Problem-solving Methods
In the next section, let’s discuss the 5-stage strategy a little further. First, what do you consider to be the correct order for the 5-stages?
Applying the 5-Stage Strategy
The table below gives an outline of the 5-stage strategy to solving physics problems. Each stage is then expanded on by using the Cold Run problem as an example.
| Stages of Solving a Physics Problem |
| 1. Extract key elements from the problem: Extract the known and unknown information, organizing and displaying these in a coherent diagram or description. The relevant information may appear in the problem in any of the modes: language, figures, and symbolism. |
| 2. Interpret the problem using a physics model: Interpret the problem elements and relationships in terms of known physics terms and models. This stage requires a detailed analysis of the problem in combination with previously learned physics models. |
| 3. Operationalize a Solution Mathematically: Based on models selected, identify the mathematical equations and methods needed to solve for unknown elements. This stage is mathematically-focused, requiring defined symbols and quantities from the problem. |
| 4. Solve the problem: Apply the selected equations and methods, plugging in known values and resolving unknown values. Solving requires clearly defined mathematical symbols and logic in coordination with understanding the physical aspects of the problem. |
| 5. Check you answer: Apply alternative methods to check numerical solutions for physical plausibility and algebraic solutions for limiting cases and trends. Figures and language can help visualize the physical solution to determine the validity of the original solution. |
Below we will walk through an example of applying the 5-stage strategy using the Cold Run problem.
Stage 1: Draw out key elements
The Choice Create a list of known and unknown variables.
The Why It is up to the learner to determine which quantities are ‘known’ and which are ‘unknown’. This stage is facilitated by the interpretation strategies listed in 1.1.1.
- Known:
- Distance total, [latex]d_{t}=10\;\mathrm{km}[/latex]
- Velocity of friend, [latex]v_{f}=3.0\;\mathrm{m/s}[/latex]
- My velocity, [latex]v_{m}=3.5\;\mathrm{m/s}[/latex]
- Time difference between my start and my friend’s start: [latex]\Delta t=5\;\mathrm{mins}=300\;\mathrm{s}[/latex]
- Unknown:
- Distance my friend travelled, [latex]d_{f}[/latex]
- Distance I travelled, [latex]d_{m}[/latex]
- Time my friend spent running, [latex]t_{f}[/latex]
- My time spent running, [latex]t_{m}[/latex]
Stage 2: Interpret the problem using a physics model
The Choice This is a problem of linear motion in one direction. Assuming acceleration is constant, ([latex]a=0[/latex]), our classic mechanic’s 1D motion physics model (and equations) will be relevant to this problem.
The Why Stage 1 listed the variables of velocity, time, and distance; while the question implies that our motion is in one direction with no stops. 1D motion equations are frequently used for problems containing motion in one direction and containing all three variables we listed. Therefore we will use the 1D motion equations to provide a framework for solving this problem.
Stage 3: Operationalize a Solution Mathematically
The Choice We will use 1D motion equation [latex]d=vt[/latex] to solve the problem.
The Why We have identified the problem will need 1D motion equations to solve. Analyzing our variables in stage 1, there is only one equation relevant to 1D motion that includes constant acceleration:
[latex]d=vt.[/latex]
Stage 4: Solve the Problem
The Choice We will apply the equation [latex]d=vt.[/latex] for both my motion and my friend’s motion. We will also use the condition [latex]\Delta t + t_m= t_f[/latex]. We will solve the equation to obtain the unknown [latex]t_f[/latex], [latex]t_m[/latex], and subsequently [latex]d_f[/latex] and [latex]d_m[/latex] via [latex]d=vt[/latex].
The Why We can now create two equations and insert the known values for the velocity, [latex]v_{f}[/latex] and [latex]v_{m}[/latex]. This will leaves us with two unknown variables to solve for: [latex]t[/latex] and [latex]d[/latex]. Using the previously defined term of [latex]d_{m}=d_{f}[/latex] we can eliminate the [latex]d[/latex] variable through substitution. Finally, using the equation [latex]t+300s=t[/latex] we can use a single variable [latex]t[/latex] to represent time.
The Action Combining the equations of motion of my friend and me, we have
[latex]v_f(t_m+\Delta t)=v_mt_m,[/latex]
which solves
[latex]t_m=\frac{v_f\Delta t}{v_m-v_f}.[/latex]
We plug in the number from the problem and evaluate [latex]t_m=1800\;\mathrm{s}=6\;\mathrm{min}[/latex]. By [latex]d=v_mt_m[/latex], we find [latex]d=6300\;\mathrm{m}[/latex].
Stage 5: Check your solution
The Choice Use our friend’s velocity equation to solve for the distance.
The Why Since my distance travelled and my friend‘s distance travelled are equal, we should get the same answer from both equations. By plug the number into
[latex]d=v_f(t_f+\Delta t),[/latex]
we obtained [latex]d=6300\;\mathrm{m}[/latex], which is the same as the distance I travelled.
Furthermore, our run is only 10km long and we will meet at 6.3km. This distance is a reasonable answer as it is within the range of our run! It is important to understand that this stage will be different depending on the question asked. It will take some creative problem solving to determine the best method in checking your solution.
Task 1.5: Reflecting on Strategies
The quiz below will test your knowledge and provide feedback on the 5-stage strategy. After completing the quiz, reflect on the similarities and differences between your solution in task 1.1.2 and the formal solution shared above.
Three Meaning-making Modes in Physics
Let’s discuss the three communicative tools, called modes, that we use in combination in physics: language, figures, and symbolism. It is often possible to express an idea such as “exponential growth” using any of these modes (see the task below); however, as you probably know, each of these modes specializes in particular ways of making meaning. What are the key functions of language, figures, and symbolism? In other words, what does each mode allow us to do especially well?
- Language: Language is the glue that holds meaning together in science and, indeed, in most kinds of human interaction. Through the vast choices of vocabulary and grammar that language affords us, we are able to represent the world and human experiences in highly detailed and subtle ways. For example, using language it’s possible to describe the very particular ideas and activities involved in an extended scientific procedure. Such a description would be more limited if we used just figures or mathematical symbols.
- Figures: Figures are good at is representing multiple entities and the relationships between them in space. Humans interpret space as meaning, like a line that moves from left to right often signals movement in time. The key point is that by representing multiple entities and their relationships in space, these meanings are available to be interpreted all at once in a non-linear frame.
- Symbolism: Symbolism encompasses symbols for entities such as H for height, symbols for specific kinds of logical operations, and equations for organizing such calculations. Strict rules are applied when communicating via symbols.
Task 1.6: The Three Meaning-Making Modes
The same information below is represented in three different ways. After completing the task, review the feedback on your answer.
Task 1.7: Combining Language, Figures, and Symbols
You will find an example of each of the three meaning-making modes in the task below. Click and drag each example into their corresponding mode.
Task 1.8: Figures and their Functions
In the task below you will find image’s of 5 frequently used figures in physics. These figures are particularly important for stages 2, 3 and 5 of the 5-stage strategy. Complete the task below to introduce yourself with these various figures. Click and drag the appropriate description of each figure into its corresponding box.
Task 1.9: Two Types of Figures
In physics, figures can be broadly classified into two categories: qualitative and quantitative. Qualitative figures focus on the relative relationships between entities within a diagram, while quantitative figures depict objects and entities with mathematical accuracy. The following tasks can help you understand the differences between these two types of figures:
Task 1.10: Types of Equations and Their Functions
Physics solutions typically employ three types of equations: definition, theorem, and derived equations. Defining equations give you a definition of a new quantity that is consistent across the solution (e.g. [latex]v=\frac{dx}{dt}[/latex]). Theorem are fundamental in physics and define a universal law; these equations can only be verified through rigorous experimentation (e.g. Newton’s second law for mechanics: [latex]F=ma[/latex]). Defining and theorem equations are typically found in stage 2 and 3 of the problem solving method where they aid in interpreting and operationalizing the solution mathematically. Finally, derived equations are a variant group generated uniquely for each solution by combining definition equations and theorems (e.g. [latex]2ax=v_2^2-v_1^2[/latex]). Derived equations are typically found in stage 4 of the problem solving method. At this stage they are used to determine the mathematical answer to the problem.
The following task presents a solution to a physics problem. Each step may use different equations. Use your understanding of the three equation types to determine which equation type are used within each stage.
The Problem:
A car begins driving from a stationary position. It accelerates at [latex]8.0\;\mathrm{m/s^2}[/latex] for [latex]15\;\mathrm{s}[/latex], then travels at a steady pace for another [latex]15\;\mathrm{s}[/latex], all in the same direction. How much distance has it covered since traveling?
Stage 1:
Known:
- [latex]d_{t}=10\;\mathrm{km}[/latex]
- [latex]a=8.0\;\mathrm{m/s^2}[/latex]
- [latex]t_1=15\;\mathrm{s}[/latex]
- [latex]t_2=15\;\mathrm{s}[/latex]
Unkown:
- [latex]d_t=?\;\mathrm{m}[/latex]
Stage 2:
This is a problem of linear motion in one direction. Acceleration is constant, [latex]a=5\;\mathrm{m/s^2}[/latex], for the first 12-seconds, then constant at [latex]a=0\;\mathrm{m/s^2}[/latex], for the remainder 12-seconds. Our classic mechanic’s 1D motion physics model (and equations) will be relevant to this problem. First,
[latex]d=vt[/latex]
will solve for the distance travelled when when acceleration is 0. Next, the equation
[latex]a=\frac{\Delta v}{\Delta t}[/latex]
will be needed to solve for the final velocity in the second half of the car’s travel. Then we will use the kinematic equation
[latex]d=v_it+\frac{1}{2}at^2,[/latex]
to determine our distance while accelerating at [latex]a=5\;\mathrm{m/s^2}[/latex]. Finally we will use the equation
[latex]d_t=d_i+d_f[/latex]
to determine the total distance travelled.
SECTION 1.4: Units and Scales of Meaning-making in Language
Task 1.11: Thinking Ahead about Units and Scales of Language
Language is a complex resource. A key feature of language is that it is organized into meaning-making units; just as mathematics has smaller units like numbers and variables, and larger-scale units like equations, in language words are smaller units that are organized at increasingly larger scales in sentences and whole texts. When working consciously with language choices in real-world situations such as solving physics problems, it’s important to recognize and focus on the relevant units of language units and the relationship between the units. This is the focus of the following task.
Read and order the language units from small to large scale. The following sentence may help: choose one unit for each gap.
The smallest unit here is a __1__, which combines with others to form a __2__, which typically joins with other phrases to form a __3__,
which is also a __4__ or just a of part one, which very often combines with others to form one functional step or __5__ in the message achieving its overall purpose.
Multiple stages typically combine in a whole __6__, which exemplifies one or more type of text or __7__.
Task 1.12: Genre as a Unit of Physics Culture
Genre is the term we use to describe and classify a cultural pattern or convention involving the exchange of meaning. For example, as members of contemporary global culture, we are likely to recognize different genres of music (such as hip-hop, K-pop, Eurobeat, videogame soundtrack) or films (such as documentaries, action films, love stories, and sci-fi).
The field of physics is a global scientific practice that involves many genres, including spoken genres such as the lecture, lab demonstration, tutorial dialogue, and problem-solving in student groups, as well as written ones such as the lab report, research report, and written solutions to problems. Each of these genres fulfills a different set of purposes in the culture of physics.
Read texts 1-4 below (you have already have seen text 1). All four texts are similar in important ways, even if the topics involved in each are very different; however, one of the texts is sufficiently different from the others in its structure and purpose to be classified differently from the other three texts. In other words, three of these texts may be classified as examples of a single genre – the genre of an undergraduate-level physics word problem – while one does not belong to the genre. Identify this outlier.
Task 1.13: Stage as a Unit within Genres and Texts
Clearly, texts 1-3 above have the same purposes and share similar features. As such, they are examples of the genre of physics problem. The physics problem genre varies greatly; however, physics problems contain two stages: (1) the physical setting stage, which provides the necessary background information (and sometimes unnecessary and distracting information!) and (2) the task stage, which may also contain key information but, most importantly, contains a question (e.g., “What is the… ?”) or a command (e.g., “Determine the….”).
The question is the obligatory stage of physics problems that text 4 doesn’t have. A stage is an identifiable, purposeful step (whether conscious or not) that contributes to the overall purpose of a text.
You will recall our recommendation of a 5-stage strategy for solving physics word problems where each of the stages contributes uniquely to the effectiveness of the solution. Are you able to recall the 5 stages and their individual purposes?
Stages are a key way to identify genres. We’ve discussed the staging of physics solutions; another example of staging in post-secondary education separate from physics problems and solutions is the argumentative essay, a genre that involves an Introduction stage with a thesis statement, followed by a Supporting Evidence stage, and a Conclusion stage, where the thesis is restated. As with the stages of physics problems and solutions, each stage is necessary if the text is to be effective.
Although text 4 above is not a physics problem because it lacks a task stage, it is clearly related to physics culture. The approach to genre taken in this textbook has to cope with more complex, mixed-genre texts like this that arise in physics and indeed all cultures. This point raises the next question: in which of the following genres would you classify text 4?
Task 1.14: Using Staging to Help Check for Distractors in Problems
As you probably know, physics instructors sometimes include distractors in the problem, information that is not necessary to solve the problem. They do this in order to develop your critical abilities to assess all the information in problems towards your solution. While distractors can occur anywhere in a problem, they are more likely to occur in one of the stages; this point is the focus of this task.
The Trolley Problem is broken down below by its physical setting and task stages. As shown in the four yellow highlighted phrases, both these stages can contain assumptions and quantities for solving the problem. For now, we are not concerned with solving this problem or even considering the specific highlighted values; our interest is in comparing the reliability of the background information provided in the physical setting stage versus in the task stage.
Physical Setting
Task 1.15: Identifying Units of Language Use in Examples
Within each task, choose the text that best matches the prompt given.
Task 1.16: Words, Phrases, Clauses, Sentences and the (Re)Distribution of Meaning Among Them
The table below shows how a very similar statement (in this case, a statement explaining the use of a kinematic equation) can be expressed in different ways (rows A – C) depending on the way the ideas and logical connections are distributed among various units of language. The increase in shading of each row reflects the increase in the density of information per grammatical unit; we will explore the many uses of this variation in information density in subsequent units. Review the table and complete the tasks below.
Task 1.4.6a Instructions: Fill in the blanks from the following four choices. Type your choices in the blanks to complete the statement describing the table:
Task 1.4.6b Instructions: If the information in rows A-C changes only in terms of its density, you should be able to identify how ideas are expressed differently, using different units of language, between the rows. Type in the information: spelling and spacing matter!
Task 1.4.7 (Optional) Instructions: Attempt to pack the explanation about using the kinematic equation into an informationally dense noun phrase, as would appear in row D.
End of Chapter Game
Congratulations! You have completed the first chapter. Below is a game designed where you will have to apply basic physics knowledge while answer questions about this chapter.
Can you use your new knowledge and basic physics concepts to achieve a high score?
A system of communication to detail correspondence of our real world,
Represent a relationship spatially where the different variables and quantities are usually immediately visible to the learner.
A variable that has been previously defined to mean something else (example: h=height)
Will highlight which portions of the problem are up to the student's discretion to decide
Will highlight the rationale for picking the choice
A mathematical process taken to achieve the answer to the problem
Represent a relationship spatially where the different variables and quantities are usually immediately visible to the learner.
