Challenge Activities

Section 1.13 Challenge Activities

The last section in each chapter includes a set of challenge activities. Each activity is typically either an Explanation Task, focused on conceptual understanding and using scientific reasoning, or an A*R*C*S Activity, focused on using representational fluency, symbolic algebra, and physics sensemaking to explore situations with deep physics.

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Subsubsection Challenge Activity Philosophy

When you work on a challenge activity, your goal is not just to solve a problem, but to learn new physics and practice communicating.

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Subsubsection Explanation Task Instructions

Physics is a systematic way of looking at the world that involves building quantitative models and mechanistic explanations for why real-world objects and systems behave the way they do. When you provide an explanation, include the steps detailed below.

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Explanations in Physics. 1. Principles - what fundamental physics concepts, laws, or definitions did you start with? 2. Reasoning - explain all the reasoning steps to go from your principles to your conclusion. 3. Conclusion - state your conclusion clearly. — Figure 1.13.1. Steps to follow when giving an explanation.
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Example 1.13.2. Rectangular Park I.

You manage parks for Corvallis, and you are comparing two rectangular parks that have the same total area. Park 1 is 2 km long, while park 2 is 3 km long. Is the width of park 2 greater, less than, or equal to the width of park 1?

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Principles: If \(l\) is the length and \(w\) is the width of a rectangular park, the area is \(A_{rectangle} = lw\text{.}\)

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Reasoning: The area of the two parks is constant, so we can say that \(A_1 = A_2\) or alternatively \(l_1 w_1 = l_2 w_2\text{.}\) Since \(l_2\) is greater than \(l_1\text{,}\) we must have that \(w_2\) is smaller than \(w_1\) to keep the product of l and w the same for the two parks. The width of park 2 is less than the width of park 1. A way of making sense of this is to draw pictures of two example parks that have different widths but the same area.

Figure 1.13.3. Two rectangles, one with greater length than the other.
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Conclusion: The width of park 2 is less than the width of park 1.

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Commentary

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Principles: The Principle for this problem is relatively simple! Sometimes you will only have one simple Principle like in this problem, while other times you may have several.

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Reasoning: An important claim here is that the parks have the same area. Note that even though this is stated in the problem, make sure you list it here—often there is information in the problem statement that is not directly relevant or necessary to consider. You should go through every step of the reasoning here, starting from the principle: first, from the claim that two areas are equal, use proportional reasoning to go from the given relationship between \(l\)’s to the desired relationship between \(w\)’s. You can mix words, symbols, and even pictures in your reasoning!

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Conclusion: Clearly state the conclusion of your reasoning that answers the question.

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Explanation 1.13.1. A Walk in the Park.

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You have been asked to walk your dog around a large park. Your dog always walks north first, then turns and walks in a different direction for the same distance, and last she turns again and walks in another different direction for the same distance. At the end of today’s walk, you find yourself back at the exact position where you started! Give an explanation for how this is possible using vectors.

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Tip: Draw a diagram to support your reasoning.

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Subsubsection ARC*S Activity Instructions

At this point, you have learned a wide range of skills that you can apply to a physics context, including:

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Identifying assumptions, as shown in the Assumptions Steps
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Drawing graphs and other representations, such as a Motion Graph
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Calculating an unknown quantity, as shown in the Calculation Steps
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Checking units to make sense of an answer, as described in Check Units
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Making sense of a symbolic equation using covariation, as described in Covariational Reasoning
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While each of these skills is useful on its own, they are most powerful when used together to help you examine complicated contexts that often draw on real-world experiences. Activities where you should carry out all of the steps above are labeled A*R*C*S: Analyze-Represent-Calculate-Sensemake. The figure below shows all the individual steps to help you keep track of them.

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Figure 1.13.4. Steps to follow when you consider an A*R*C*S activity.
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Example 1.13.5. ARC*S Example: Rectangular Park II.

You manage parks for Corvallis, and you are responsible for putting up fences around two new parks that have the same area. Park 1 is a square, while park 2 is a rectangle whose length is \(110 \mathrm{~m}\text{.}\) If park 2 requires twice as much fencing as park 1, what is the area of each park?

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1. Analyze and Represent

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List quantities. The only known here is \(l = 110 \mathrm{~m}\text{,}\) the length of the rectangular park. There are many unknowns: \(s\) (the side length of the square park), \(w\) (the width of the rectangle), \(P_{square}\text{,}\) \(A_{rectangle}\text{,}\) and \(P_{rectangle}\text{,}\) and \(A_{square}\) (which you are trying to find).
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Identify assumptions. There is no physics in this activity, but the problem stated that the parks are rectangular, which is probably a simplification given that real parks are often strange shapes.
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Represent the situation physically. Below is a diagram of the square and the rectangle, with all distances labeled.
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Figure 1.13.6. A rectangular park and a square park.
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2. Calculate

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Represent principles symbolically. The area of a square is \(A_{square} = s^2\) and the perimeter (the length of the fence) is \(P_{square} = 4s\text{.}\) The area of a rectangle is \(A_{rectangle} = lw\) and the perimeter (the length of the fence) is \(P_{rectangle} = 2l + 2w\text{.}\)
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Solve unknown(s) symbolically. Start by setting the two areas equal and relating the perimeters

\begin{equation*} A_{square} = A_{rectangle} \end{equation*}

\begin{equation*} s^2 = lw \end{equation*}

\begin{equation*} P_{rectangle} = 2P_{square} \end{equation*}

\begin{equation*} 2l + 2w = 8s \end{equation*}

Combine these expressions algebraically by solving the first equation for \(w\) and plugging it into the second equation:

\begin{equation*} w = s^2/l \end{equation*}

\begin{equation*} 2l + 2s^2/l = 8s \end{equation*}

\begin{equation*} s^2 - 4sl + l^2 = 0 \end{equation*}

Use the quadratic formula to solve this last equation for \(s\text{,}\) then square \(s\) to get the area of the square!

\begin{equation*} s = \frac{4l \pm \sqrt{16l^2 - 4l^2}}{2} = \left(2 \pm \sqrt{3}\right)l \end{equation*}

\begin{equation*} A = s^2 = \left(7 \pm 4\sqrt{3}\right)l^2 \end{equation*}

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Plug in numerical values. The two possible approximate values for \(A\) are \(13.9l^2 \approx 168190 \mathrm{~m^2}\) or \(0.07l^2 \approx 847 \mathrm{~m^2}\text{.}\)
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3. Sensemake

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Are the units correct? The units for area should be \(\mathrm{m^2}\text{.}\) Each factor of \(l\) in the answer has units of \(\mathrm{m}\text{,}\) so squaring it gives \(\mathrm{m^2}\text{,}\) which is what you should expect!
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Is your numerical answer reasonable? Something a little weird happened. You got two numbers. What does that mean? In this case you were told that the length of the park was \(l\text{,}\) but you were not told whether the length was bigger or smaller than the width! The two cases correspond to those two possibilities. The units \(\mathrm{m^2}\) are not great for making sense of something as big as a park, so you can look up how many \(\mathrm{m^2}\) are in an acre (4047), which is a more reasonable measure of land area for a park. This gives 41.6 acres (this would be a pretty big park!) and 0.2 acres (on the other hand, very small).
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Does your symbolic answer make physical sense? A quick way to check if your answer makes sense is to evaluate how it depends on the given quantities. In this case, the area varies like \(l^2\text{.}\) First of all, it depends on the square of \(l\text{,}\) which is something you have seen before in many other expressions for area, such as for squares, circles, polygons, etc. More specifically, this means that increasing \(l\) (making the given parameter bigger) corresponds to an increase in the area. Lastly, you can check special values of \(l\) and see what your answer is. The most special value of \(l\) is \(l = 0\text{,}\) which corresponds to no park at all! In this case, the area of the park should of course be 0, which is what your symbolic answer says as well:

\begin{equation*} A = s^2 = \left(7 \pm 4\sqrt{3}\right) 0^2 = 0 \end{equation*}

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ARC*S 1.13.2. The Tower in the Forest.

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You and a friend are each lost in the same forest. They message you that they are directly northeast of a very large tower that is about \(3\) miles away. You are about \(2\) miles away from the same large tower, but you are \(25^o\) west of south of it. About how far are you from your friend? What direction would you need to walk to reach your friend?

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