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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 1.14 Challenge - Vectors

Subsubsection Explanation Tasks

For each Explanation Task, you are expected to complete and label all of the Explanation Task Steps. A simple example is provided in Rectangular Park I.
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Explanation 1.14.1. Subtracting Equal Vectors.

In each case below, the top vector is being subtracted from the bottom vector. All of the vectors have the same magnitude. Is the magnitude of the difference in the left case greater than, less than, or equal to the magnitude of the difference in the right case?
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Tip: Draw a diagram to support your reasoning.
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Figure 1.14.1. Two pairs of vectors.
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Subsubsection A*R*C*S Activities

For each A*R*C*S activity, you are expected to complete and label all of the A*R*C*S Steps. For your first A*R*C*S activity, each step will walk you through the A*R*C*S process. A simple example is provided in A*R*C*S Example: Rectangular Park II.
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A*R*C*S 1.14.2. The Trees upon the Hillside.

A surveyor measures a hillside and describes the height of the hill \(y\) (in meters) as the following function of distance from the base of the hill \(x\) (also measured in meters):
\begin{equation*} y(x) = a_1x^3 - a_2x^5 \end{equation*}
He reports that \(a_1 = 1.5 \times 10^{-5}\) and \(a_2 = 1.0 \times 10^{-10}\text{.}\) He has also identified three especially large trees (about \(30 \mathrm{~m}\) tall) whose bases are located at \(x_1 = 5 \mathrm{~m}\text{,}\) \(x_2 = 257 \mathrm{~m}\text{,}\) and \(x_3 = 300 \mathrm{~m}\text{.}\)
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1. Analyze and Represent
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  1. Identify known and unknown quantities with both a symbol and a number. In general, when a quantity is stated with a number of symbol, that is a known quantity.
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    Tip: It looks like the surveyor forgot to report the specific units for the constants \(a_1\) and \(a_2\text{.}\) Determine appropriate units for these constants. Describe why these are sensible units given the equation for \(y(x)\text{.}\)
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  2. Identify and justify any assumptions.
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  3. Sketch and label a quantitatively accurate diagram of the hillside and the trees.
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    Tip: Make sure to identify your origin, label your axes, and identify any relevant points of interest.
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2. Calculate
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  1. Represent physics principles that will help you solve for the unknown quantities (you only really need one here!).
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  2. Determine a symbolic expression for the displacement vector from the base of tree 2 to the base of tree 3 in terms of the given variables.
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  3. Plug numbers into your symbolic answer.
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3. Sensemake
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  1. Check the units of your symbolic answer.
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  2. Compare your numerical answers to other numbers in the situation. Find at least one other distance to compare it to: should it be bigger or smaller than that other distance? Explain your reasoning.
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  3. What symbols does your numerical answer depend on? In what ways does it depend on each symbol? Why do you think this does (or does not) make physical sense, given the situation?
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Subsubsection Metacognitive Reflection

Choose at least one of the Challenge Activities. Reflect on how this activity was similar to or different from other activities you worked on throughout this chapter (these can be in or out of class activities). Cite specific activities by name and be specific about how the solutions, not just the activities themselves, were related. Include a sentence at the end about why you think a reflection activity like this might be useful in helping you learn physics.
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