1.5: Historical Context
Context of Creation
The political, socioeconomic, and cultural setting that a work of art is created in will affect how it is perceived within art history.
Recognize the importance of an artwork’s context of creation to art history
Key Points
- Patronage of the arts, and art history by extension, has been used throughout history to endorse the ambitions and agenda of the dominant power of any given age. Art history is the academic study of objects of art in their historical development and stylistic contexts (i.e., genre , design, form , and style ).
- Art conveys political, religious, and philosophical themes and judgments that arise as much from the artist’s environment as they do from his or her creative impulse.
- Some of the contextual forces that shape artists and their work are their teachers and the influences of preceding styles; their patrons and their demands; their audiences; and their general socioeconomic, political, and cultural climate.
Key Terms
- iconography :The branch of art history which studies the identification, description, and the interpretation of the content of images.
- oeuvre :The complete body of an artist’s work.
Art has existed almost as long as humankind itself and serves as a vehicle for the expression and communication of ideas and emotions. The canon of art history, however, has historically conveyed the political, religious, and philosophical ideals of the dominant power. Art history categorizes artworks and theories with a heavy reliance on the context or environment that the artwork was created in (i.e., its political, social, cultural, and economic settings).
Art history is the academic study of art objects in their historical development and stylistic contexts (i.e., genre, design, form, and style). A work of art from a particular historical period can be treated as an original source of information that was created at the time under study, and provides information about that time. Art historians study the contextual forces that shaped artists and their oeuvres , including their teachers and the influences of preceding styles; their patrons and their demands; their audiences; and their general socioeconomic, political, and cultural climate. These factors produce and influence different artistic styles and iconography , which are characteristic of their age and geographical location with reference to visual appearance, technique, and form.
In many ways, the historical backbone of art history is a celebratory chronology of beautiful creations of art commissioned by religious or civic institutions or wealthy individuals. Patronage of the arts has been used throughout history to endorse the ambitions and agenda of these institutions and individuals, and has been particularly important in the creation of religious art . For example, the Roman Catholic Church was an enthusiastic sponsor of the arts that resulted in a tremendous outpouring of architecture, painting, sculpture , and decorative crafts in medieval and Renaissance Europe.
Intended Context of Reception
Art’s context of reception depends on a variety of circumstances, both on the part of the artist as well as the artistic community the artist is participating in.
Identify the non-motivated, as well as motivated, factors that have given rise to art
Key Points
- Art arises from a combination of non-motivated factors driven by the intrinsic human impulse towards harmony and creativity as well as motivated factors, which consciously aim to communicate specific messages to other individuals.
- Art may be used to evoke particular emotions or moods, for social inquiry and political change, for questioning and criticizing society, or as a means of propaganda or commercial advertisement for influencing popular conceptions.
- Religious art uses religious inspiration and themes in order to illustrate the principles of the religion and to provide spiritual instruction to audiences.
- Patronage of the arts was typically used as a means of expressing and endorsing political, social, and cultural agendas and of displaying personal prestige. Works of art commissioned by wealthy patrons usually reflect their desires and aims.
Key Terms
- patron :An influential, wealthy person who supports an artist, craftsman, scholar, or aristocrat.
- motif :A recurring or dominant element in a work of art.
Art’s context of reception depends on a variety of circumstances, both on the part of the artist as well as the artistic community and climate that the artist is participating in. Throughout human history, art has been created across a range of media for many different reasons and to serve many different functions. Some of these purposes are intrinsic to the human instinct for harmony and balance, as well as the human desire to experience mysterious things and express the human imagination. Art can transcend the concept of utility or external purpose. These ideas are called the non-motivated purposes of art. However, art also comes from intentional, conscious actions that aim towards specific external goals, and those qualify as the motivated purposes of art. Motivated purposes usually arise from the artwork’s historical context, which consists of a multitude of different factors, including the social, political, economic, and cultural settings of the period; the artist’s patrons; and the artist’s intended audience.
Primarily, art is a form of communication, and like most forms of communication, has intents and goals directed toward other people. It may be used for entertainment, seeking to evoke particular emotions or moods in viewers , or for social inquiry and political change by portraying aspects of society in complimentary or critical ways.
Similarly, art may also be used as a form of propaganda by subtly influencing popular conceptions, or for commercial purposes, by making specific products more attractive to potential consumers. Religious or sacred art uses religious inspiration and motifs in order to illustrate the principles of a religion in a tangible form, and is often intended to provide spiritual instruction and connection with believers.
Through the course of history, much of art has traditionally been patronized by wealthy and powerful individuals, including rulers and aristocrats, as well as various civic and religious institutions. Patronage of the arts was typically used as a means of expressing and endorsing political, social, and cultural agendas and of displaying personal prestige. Works of art commissioned by wealthy patrons usually reflect their desires and aims.
- Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Creaciu00f3n de Adu00e1n (Miguel u00c1ngel). Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Creaci%C3%B3n_de_Ad%C3%A1n_(Miguel_%C3%81ngel).jpg. License : Public Domain: No Known Copyright
- The Creation of Adam. Provided by : Wikipedia . Located at : en.Wikipedia.org/wiki/The_Creation_of_Adam. License : CC BY-SA: Attribution-ShareAlike
- Art. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Art. License : CC BY-SA: Attribution-ShareAlike
- Patronage. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Patronage. License : CC BY-SA: Attribution-ShareAlike
- Primary source. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Primary_source. License : CC BY-SA: Attribution-ShareAlike
- Art history. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Art_history. License : CC BY-SA: Attribution-ShareAlike
- iconography. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/iconography. License : CC BY-SA: Attribution-ShareAlike
- oeuvre. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/oeuvre. License : CC BY-SA: Attribution-ShareAlike
- Eugu00e8ne Delacroix - La libertu00e9 guidant le peuple. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Eug%C3%A8ne_Delacroix_-_La_libert%C3%A9_guidant_le_peuple.jpg. License : Public Domain: No Known Copyright
- Sandro Botticelli 061. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/File:Sandro_Botticelli_061.jpg. License : Public Domain: No Known Copyright
- Patronage. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Patronage. License : CC BY-SA: Attribution-ShareAlike
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- patron. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/patron. License : CC BY-SA: Attribution-ShareAlike
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4.3: Annuities and Sinking Funds
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Learning Objectives
In this section, you will learn to:
- Find the future value of an annuity.
- Find the amount of payments to a sinking fund.
Ordinary Annuity
In the first two sections of this chapter, we examined problems where an amount of money was deposited lump sum in an account and was left there for the entire time period. Now we will do problems where timely payments are made in an account. When a sequence of payments of some fixed amount are made in an account at equal intervals of time, we call that an annuity . And this is the subject of this section.
To develop a formula to find the value of an annuity, we will need to recall the formula for the sum of a geometric series. A geometric series is of the form:
\[a + ax + ax^2 + ax^3+ \ldots + ax^n. \label{eq1} \]
In a geometric series, each subsequent term is obtained by multiplying the preceding term by a number, called the common ratio . A geometric series is completely determined by knowing its first term, the common ratio, and the number of terms.
The first term of the series in Equation \ref{eq1} is \(a\), the common ratio is \(x\), and the number of terms is \(n+1\). The following are some examples of geometric series.
\[3 + 6 + 12 + 24 + 48 \nonumber \]
This above series has first term \(a = 3\) and common ratio \(x = 2\)
\[2 + 6 + 18 + 54 + 162\nonumber \]
This above series has first term \(a = 2\) and common ratio \(x = 3\)
\[37 + 3.7 + 0.37 + 0.037 + 0.0037\nonumber \]
This above series has first term \(a = 35\) and common ratio \(x = 0.1\)
In your algebra class, you developed a formula for finding the sum of a geometric series. You probably used \(r\) as the symbol for the ratio, but we are using \(x\) because \(r\) is the symbol we have been using for the interest rate. The formula for the sum of the terms of a geometric series with first term \(a\), common ratio \(x\) and last term \(ax^{n}\) is:
\[\frac{a\left(x^{n+1}-1\right)}{x-1} \nonumber \]
We will use this formula to find the value of an annuity. Consider the following example.
Example \(\PageIndex{1}\)
If at the end of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?
Solution
There are 60 deposits made in this account. The first payment stays in the account for 59 months, the second payment for 58 months, the third for 57 months, and so on.
- The first payment of $500 will accumulate to an amount of $500(1 + 0.08/12) 59 .
- The second payment of $500 will accumulate to an amount of $500(1 + 0.08/12) 58 .
- The third payment will accumulate to $500(1 + 0.08/12) 57 .
- The fourth payment will accumulate to $500(1 + 0.08/12) 56 .
And so on . . .
Finally the next to last (59 th ) payment will accumulate to \($500(1 + 0.08/12)^1\).
The last payment is taken out the same time it is made, and will not earn any interest.
To find the total amount in five years, we need to add the accumulated value of these sixty payments.
In other words, we need to find the sum of the following series.
\[\$ 500(1+0.08 / 12)^{59}+\$ 500(1+0.08 / 12)^{58}+\$ 500(1+0.08 / 12)^{57}+\ldots+\$ 500 \nonumber \]
Written backwards, we have
\[ \$500 +\$ 500(1+0.08 / 12)+\$ 500(1+0.08 / 12)^{2}+\ldots+\$ 500(1+0.08 / 12)^{59} \nonumber \]
This is a geometric series with \(a = $500\), \(r = (1 + 0.08/12\), and \(n = 59\). The sum is
\[\begin{align*} \text{sum} &= \dfrac{\$ 500\left[(1+0.08 / 12)^{60}-1\right]}{0.08 / 12} \\[4pt] &=\$ 500(73.47686) \\[4pt] &=\$ 36,738.43 \end{align*} \nonumber \]
When the payments are made at the end of each period rather than at the beginning, we call it an ordinary annuity .
Future Value of an Ordinary Annuity
If a payment of m dollars is made in an account n times a year at an interest \(r\), then the final amount \(A\) after \(t\) years is
\[\mathbf{A}=\frac{\mathbf{m}\left[(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n} \mathbf{t}}-\mathbf{1}\right]}{\mathbf{r} / \mathbf{n}} \nonumber \]
The future value is also called the accumulated value
Example \(\PageIndex{2}\)
Tanya deposits $300 at the end of each quarter in her savings account. If the account earns 5.75% compounded quarterly, how much money will she have in 4 years?
Solution
The future value of this annuity can be found using the above formula.
\[\begin{align*} A&=\frac{\$ 300\left[(1+.0575 / 4)^{16}-1\right]}{0.0575 / 4} \\[4pt] &=\$ 300(17.8463) \\[4pt] &=\$ 5353.89 \end{align*} \nonumber \]
If Tanya deposits $300 into a savings account earning 5.75% compounded quarterly for 4 years, then at the end of 4 years she will have $5,353.89
Example \(\PageIndex{3}\)
Robert needs $5,000 in three years. How much should he deposit each month in an account that pays 8% compounded monthly in order to achieve his goal?
Solution
If Robert saves m dollars per month, after three years he will have
\[\dfrac{m\left[(1+.08 / 12)^{36}-1\right]}{0.08 / 12} \nonumber \]
But we'd like this amount to be $5,000. Therefore,
\[\begin{align*} \frac{\mathrm{m}\left[(1+.08 / 12)^{36}-1\right]}{.08 / 12} &=\$ 5000 \\[4pt] \mathrm{m}(40.5356) &=\$ 5000 \\[4pt] \mathrm{m} &=\frac{5000}{40.5356}\\[4pt] &=\$ 123.35 \end{align*} \nonumber \]
Robert needs to deposit $123.35 at the end of each month for 3 years into an account paying 8% compounded monthly in order to have $5,000 at the end of 5 years.
Sinking Fund
When a business deposits money at regular intervals into an account in order to save for a future purchase of equipment, the savings fund is referred to as a “ sinking fund ”. Calculating the sinking fund deposit uses the same method as the previous problem.
Example \(\PageIndex{4}\)
A business needs $450,000 in five years. How much should be deposited each quarter in a sinking fund that earns 9% compounded quarterly to have this amount in five years?
Solution
Again, suppose that \(m\) dollars are deposited each quarter in the sinking fund. After five years, the future value of the fund should be $450,000. This suggests the following relationship:
\[\begin{align*} \dfrac{\mathrm{m}\left[(1+ 0.09 / 4)^{20}-1\right]}{0.09 / 4} &=\$ 450,000 \\[4pt] \mathrm{m}(24.9115) &=450,000 \\[4pt]
\mathrm{m} &=\dfrac{450000}{24.9115} \\[4pt] &=\$ 18,063.93 \end{align*} \nonumber \]
The business needs to deposit $18,063.93 at the end of each quarter for 5 years into an sinking fund earning interest of 9% compounded quarterly in order to have $450,000 at the end of 5 years.
Annuity Due
If the payment is made at the beginning of each period, rather than at the end, we call it an annuity due . The formula for the annuity due can be derived in a similar manner. Reconsider Example 1, with the change that the deposits are made at the beginning of each month.
Example \(\PageIndex{5}\)
If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?
Solution
There are 60 deposits made in this account. The first payment stays in the account for 60 months, the second payment for 59 months, the third for 58 months, and so on.
- The first payment of $500 will accumulate to an amount of \($500(1 + 0.08/12)^{60}\).
- The second payment of $500 will accumulate to an amount of \($500(1 + .08/12)^{59}\).
- The third payment will accumulate to \($500(1 + 0.08/12)^{58}\).
And so on . . .
The last payment is in the account for a month and accumulates to \($500(1 + 0.08/12)\)
To find the total amount in five years, we need to find the sum of the series:
\[$500(1 + 0.08/12)^{60} + $500(1 + 0.08/12)^{59} + $500(1 + 0.08/12)^{58} + \ldots + $500(1 + 0.08/12) \nonumber \]
Written backwards, we have
\[$500(1 + 0.08/12) + $500(1 + 0.08/12)^2 + \ldots + $500(1 + 0.08/12)^{60} \nonumber \]
If we add $500 to this series, and later subtract that $500, the value will not change. We get
\[\textbf{\$500} + $500(1 + 0.08/12) + $500(1 + 0.08/12)^2 + \ldots + $500(1 + 0.08/12)^{60} - \textbf{\$500} \nonumber \]
Except for the last term, we have a geometric series with \(a\) = $500, \(r\) = (1 + .08/12), and \(n\) = 60. Therefore the sum is
\[\begin{align*} \mathrm{A} &=\frac{\$ 500\left[(1+0.08 / 12)^{61}-1\right]}{0.08 / 12}-\$ 500 \\[4pt] &=\$ 500(74.9667)-\$ 500 \\[4pt] &=\$ 37483.35-\$ 500 \\[4pt] &=\$ 36983.35 \end{align*} \nonumber \]
So, in the case of an annuity due, to find the future value, we increase the number of periods \(n\) by 1, and subtract one payment.
Future Value of an "Annuity Due"
\[\mathrm{A}=\frac{\mathrm{m}\left[(1+\mathrm{r} / \mathrm{n})^{\mathrm{nt}+1}-1\right]}{\mathrm{r} / \mathrm{n}}-\mathrm{m} \nonumber \]
Most of the problems we are going to do in this chapter involve ordinary annuities, therefore, we will down play the significance of the last formula for the annuity due. We mentioned the formula for the annuity due only for completeness.
Summary
Finally, it is the author's wish that the student learn the concepts in a way that he or she will not have to memorize every formula. It is for this reason formulas are kept at a minimum. But before we conclude this section we will once again mention one single equation that will help us find the future value, as well as the sinking fund payment.
If a payment of \(m\) dollars is made in an account \(n\) times a year at an interest \(r\), then the future value \(\mathrm{A}\) after \(t\) years is
\[\mathbf{A}=\frac{\mathbf{m}\left[(\mathbf{1}+\mathbf{r} / \mathbf{n})^{\mathbf{n t}}-\mathbf{1} \right]}{\mathbf{r} / \mathbf{n}} \nonumber \]
Note that the formula assumes that the payment period is the same as the compounding period. If these are not the same, then this formula does not apply.


10.12: Key Takeaways and References
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Key Takeaways
- Many methods exist to quantify the timing or magnitude of an ERP component or experimental effect. The traditional methods—peak amplitude and peak latency—have many shortcomings, and better methods are available.
- In many cases, it is advantageous to obtain scores from a difference wave that isolates the component or experimental effect of interest.
- You should always visually verify that the scores are being calculated appropriately for each individual ERP waveform.
- Most scoring methods require specifying a measurement window, and this needs to be done in an unbiased manner. If you decide on the measurement window after seeing the waveforms, you may consciously or unconsciously choose a window that increases the probability of bogus-but-significant effects.
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When you have many conditions and/or channels, it’s easy to accidentally put the cells of the design into the wrong order in the statistical analysis. You should always check the table of means produced by your statistical package and make sure it matches what you are seeing in the grand average ERP waveforms.
- For mean amplitude, but not most other scoring methods, taking the score from the grand average waveforms gives you the same result as measuring from the single-participant waveforms and then averaging. This makes it easy to compare the table of means with the grand averages.
- For other measures, you can still make sure that the table of means shows the same pattern as the grand averages, even if the individual values are not identical.
- To reduce the number of p values and the likelihood of bogus-but-significant effects, you should use the smallest possible number of factors in your statistical analyses. This can often be achieved by collapsing across channels and obtaining scores from difference waves.
References
Donchin, E., & Heffley, E. F. (1978). Multivariate analysis of event-related potential data: A tutorial review. In D. Otto (Ed.), Multidisciplinary Perspectives in Event-Related Brain Potential Research (pp. 555–572). U.S. Government Printing Office.
Eriksen, C. W. (1995). The flankers task and response competition: A useful tool for investigating a variety of cognitive problems. Visual Cognition , 2 , 101–118.
Gehring, W. J., Liu, Y., Orr, J. M., & Carp, J. (2012). The error-related negativity (ERN/Ne). In S. J. Luck & E. S. Kappenman (Eds.), The Oxford Handbook of Event-Related Potential Components (pp. 231–292). Oxford University Press.
Gratton, G., Coles, M. G. H., Sirevaag, E. J., Eriksen, C. W., & Donchin, E. (1988). Pre- and post-stimulus activation of response channels: A psychophysiological analysis. Journal of Experimental Psychology: Human Perception and Performance , 14 , 331–344.
Kappenman, E. S., Farrens, J. L., Zhang, W., Stewart, A. X., & Luck, S. J. (2021). ERP CORE: An Open Resource for Human Event-Related Potential Research. NeuroImage , 225 , 117465. https://doi.org/10.1016/j.neuroimage.2020.117465
Kiesel, A., Miller, J., Jolicoeur, P., & Brisson, B. (2008). Measurement of ERP latency differences: A comparison of single-participant and jackknife-based scoring methods. Psychophysiology , 45 , 250–274. https://doi.org/10.1111/j.1469-8986.2007.00618.x
Love, J., Selker, R., Marsman, M., Jamil, T., Dropmann, D., Verhagen, J., Ly, A., Gronau, Q. F., Šmíra, M., Epskamp, S., Matzke, D., Wild, A., Knight, P., Rouder, J. N., Morey, R. D., & Wagenmakers, E.-J. (2019). JASP: Graphical Statistical Software for Common Statistical Designs. Journal of Statistical Software , 88 (1), 1–17. https://doi.org/10.18637/jss.v088.i02
Luck, S. J. (2014). An Introduction to the Event-Related Potential Technique, Second Edition . MIT Press.
Luck, S. J., & Gaspelin, N. (2017). How to get statistically significant effects in any ERP experiment (and why you shouldn’t). Psychophysiology , 54 , 146–157. https://doi.org/10.1111/psyp.12639
Luck, S. J., Stewart, A. X., Simmons, A. M., & Rhemtulla, M. (2021). Standardized measurement error: A universal metric of data quality for averaged event-related potentials. Psychophysiology , 58 , e13793. https://doi.org/10.1111/psyp.13793
Smulders, F. T. Y., & Miller, J. O. (2012). The Lateralized Readiness Potential. In S. J. Luck & E. S. Kappenman (Eds.), The Oxford Handbook of Event-Related Potential Components (pp. 209–229). Oxford University Press.
Urbach, T. P., & Kutas, M. (2002). The intractability of scaling scalp distributions to infer neuroelectric sources. Psychophysiology , 39 , 791–808. https://doi.org/10.1017/S0048577202010648
Urbach, T. P., & Kutas, M. (2006). Interpreting event-related brain potential (ERP) distributions: Implications of baseline potentials and variability with application to amplitude normalization by vector scaling. Biological Psychology , 72 (3), 333–343. https://doi.org/10.1016/j.biopsycho.2005.11.012


18.4: Motion in Two and Three Dimensions
Check Your Understanding
4.1. (a) Taking the derivative with respect to time of the position function, we have \(\vec{v}\)(t) = 9.0t 2 \(\hat{i}\) and \(\vec{v}\)(3.0s) = 81.0 \(\hat{i}\) m/s. (b) Since the velocity function is nonlinear, we suspect the average velocity is not equal to the instantaneous velocity. We check this and find$$\vec{v}_{avg} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{1})}{t_{2} - t_{1}} = \frac{\vec{r} (4.0\; s) - \vec{r} (2.0\; s)}{4.0\; s - 2.0\; s} = \frac{(188.0\; \hat{i} - 20.0\; \hat{i})\; m}{2.0\; s} = 84.0\; \hat{i}\; m/s,$$ which is different from \(\vec{v}\)(3.0 s) = 81.0 \(\hat{i}\) m/s.
4.2. The acceleration vector is constant and doesn’t change with time. If a, b, and c are not zero, then the velocity function must be linear in time. We have \(\vec{v}\)(t) = \(\int \vec{a}\)dt = \(\int\) (a \(\hat{i}\) + b \(\hat{j}\) + c \(\hat{k}\))dt = (a \(\hat{i}\) + b \(\hat{j}\) + c \(\hat{k}\))t m/s, since taking the derivative of the velocity function produces \(\vec{a}\)(t). If any of the components of the acceleration are zero, then that component of the velocity would be a constant.
4.3. (a) Choose the top of the cliff where the rock is thrown from the origin of the coordinate system. Although it is arbitrary, we typically choose time t = 0 to correspond to the origin. (b) The equation that describes the horizontal motion is x = x 0 + v x t. With x 0 = 0, this equation becomes x = v x t. (c) Equation 4.27 through Equation 4.29 and Equation 4.46 describe the vertical motion, but since y 0 = 0 and v 0y = 0, these equations simplify greatly to become y = \(\frac{1}{2}\)(v 0y + v y )t = \(\frac{1}{2}\)v y t, v y = −gt, y = − \(\frac{1}{2}\)gt 2 , and v y 2 = −2gy. (d) We use the kinematic equations to find the x and y components of the velocity at the point of impact. Using v y 2 = −2gy and noting the point of impact is −100.0 m, we find the y component of the velocity at impact is v y = 44.3 m/s. We are given the x component, v x = 15.0 m/s, so we can calculate the total velocity at impact: v = 46.8 m/s and \(\theta\) = 71.3° below the horizontal.
4.4. The golf shot at 30°.
4.5. 134.0 cm/s
4.6. Labeling subscripts for the vector equation, we have B = boat, R = river, and E = Earth. The vector equation becomes \(\vec{v}_{BE}\) = \(\vec{v}_{BR}\) + \(\vec{v}_{RE}\). We have right triangle geometry shown in the figure below. Solving for \(\vec{v}_{BE}\), we have$$v_{BE} = \sqrt{v_{BR}^{2} + v_{RE}^{2}} = \sqrt{4.5^{2} + 3.0^{2}}$$$$v_{BE} = 5.4\; m/s, \quad \theta = \tan^{-1} \left(\dfrac{3.0}{4.5}\right) = 33.7^{o} \ldotp\]
Conceptual Questions
1. Straight line
3. The slope must be zero because the velocity vector is tangent to the graph of the position function.
5. No, motions in perpendicular directions are independent.
7. a. No; b. minimum at apex of trajectory and maximum at launch and impact; c. no, velocity is a vector; d. yes, where it lands
9. They both hit the ground at the same time.
11. Yes
13. If he is going to pass the ball to another player, he needs to keep his eyes on the reference frame in which the other players on the team are located.
15.
Problems
17. \(\vec{r}\) = 1.0 \(\hat{i}\) − 4.0 \(\hat{j}\) + 6.0 \(\hat{k}\)
19. \(\Delta \vec{r}_{Total}\) = 472.0 m \(\hat{i}\) + 80.3 m \(\hat{j}\)
21. Sum of displacements = −6.4 km \(\hat{i}\) + 9.4 km \(\hat{j}\)
23. a. \(\vec{v}\)(t) = 8.0t \(\hat{i}\) + 6.0t 2 \(\hat{k}\), \(\vec{v}\)(0) = 0, \(\vec{v}\)(1.0) = 8.0 \(\hat{i}\) + 6.0 \(\hat{k}\) m/s
b. \(\vec{v}_{avg}\) = 4.0 \(\hat{i}\) + 2.0 \(\hat{k}\) m/s
25. \(\Delta \vec{r}_{1}\) = 20.00 m \(\hat{j}\), \(\Delta \vec{r}_{2}\) = (2.000 x 10 4 m) (cos 30° \(\hat{i}\) + sin 30° \(\hat{j}\)), \(\Delta \vec{r}\) = 1.700 x 10 4 m \(\hat{i}\) + 1.002 x 10 4 m \(\hat{j}\)
27. a. \(\vec{v}\)(t) = (4.0t \(\hat{i}\) + 3.0t \(\hat{j}\))m/s, \(\vec{r}\)(t) = (2.0t 2 \(\hat{i}\) + \(\frac{3}{2}\)t 2 \(\hat{j}\)) m
b. x(t) = 2.0t 2m, y(t) = \(\frac{3}{2}\)t 2 m, t 2 = \(\frac{x}{2} \Rightarrow\) y = \(\frac{3}{4}\)x
29. a. \(\vec{v}\)(t) = (6.0t \(\hat{i}\) − 21.0t 2 \(\hat{j}\) + 10.0t −3 \(\hat{k}\)) m/s
b. \(\vec{a}\)(t) = (6.0 \(\hat{i}\) − 42.0t \(\hat{j}\) − 30t −4 \(\hat{k}\)) m/s 2
c. \(\vec{v}\)(2.0s) = (12.0 \(\hat{i}\) − 84.0 \(\hat{j}\) + 1.25 \(\hat{k}\)) m/s
d. \(\vec{v}\)(1.0 s) = (6.0 \(\hat{i}\) − 21.0 \(\hat{j}\) + 10.0 \(\hat{k}\)) m/s, |\(\vec{v}\)(1.0 s)| = 24.0 m/s; \(\vec{v}\)(3.0 s) = (18.0 \(\hat{i}\) − 189.0 \(\hat{j}\) + 0.37 \(\hat{k}\)) m/s, |\(\vec{v}\)(3.0 s)| = 190.0 m/s
e. \(\vec{r}\)(t) = (3.0t 2 \(\hat{i}\) − 7.0t 3 \(\hat{j}\) − 5.0t −2 \(\hat{k}\)) cm, \(\vec{v}_{avg}\) = (9.0 \(\hat{i}\) − 49.0 \(\hat{j}\) − 6.3 \(\hat{k}\)) m/s
31. a. \(\vec{v}\)(t) = −sin(1.0t) \(\hat{i}\) + cos(1.0t) \(\hat{j}\) + \(\hat{k}\)
b. \(\vec{a}\)(t) = −cos(1.0t) \(\hat{i}\) − sin(1.0t) \(\hat{j}\)
33. a. t = 0.55 s
b. x = 110 m
35. a. t = 0.24s, d = 0.28 m
b. They aim high.
37. a. t = 12.8 s, x = 5619 m
b. v y = 125.0 m/s, v x = 439.0 m/s, |\(\vec{v}\)| = 456.0 m/s
39. a.  R Moon = 48 m
45. a. v 0y = 24 m/s, v y 2 = v 0y 2 − 2gy \(\Rightarrow\) h = 23.4 m
b. t = 3 s, v 0x = 18 m/s, x = 54 m
c. y = −100 m, y 0 = 0, y − y 0 = v 0y , t − \(\frac{1}{2}\)gt 2 − 100 = 24t − 4.9t 2 \(\Rightarrow\) t = 7.58 s
d. x = 136.44 m
e.$$\begin{split} t & = 2.0\; s, y = 28.4\; m, x = 36\; m \\ t & = 4.0\; s, y = 17.6\; m, x = 22.4\; m \\ t & = 6.0\; s, y = −32.4\; m, x = 108\; m \end{split}\]
47. v 0y = 12.9 m/s, y − y 0 = v 0y t − \(\frac{1}{2}\)gt 2 − 20.0 = 12.9t − 4.9t 2
t = 3.7 s, v 0x = 15.3 m/s \(\Rightarrow\) x = 56.7 m
So the golfer’s shot lands 13.3 m short of the green.
49. a. R = 60.8 m
b. R = 137.8 m
51. a. v y 2 = v 0y 2 − 2gy \(\Rightarrow\) y = 2.9 m/s
y = 3.3 m/s
y = \(\frac{v_{0y}^{2}}{2g}\) = \(\frac{(v_{0} \sin \theta)^{2}}{2g} \Rightarrow \sin \theta\) = 0.91 \(\Rightarrow\) \(\theta\) = 65.5°
53. R = 18.5 m
55. y = (tan\(\theta_{0}\))x − \(\Big[ \frac{g}{2(v_{0} \cos \theta_{0})^{2}} \Big]\)x 2 \(\Rightarrow\) v 0 = 16.4 m/s
57. R = \(\frac{v_{0}^{2} \sin 2 \theta_{0}}{g} \Rightarrow \theta_{0}\) = 15.0°
59. It takes the wide receiver 1.1 s to cover the last 10 m of his run.
T tof = \(\frac{2(v_{0} \sin \theta)}{g} \Rightarrow \sin \theta\) = 0.27 \(\Rightarrow \theta\) = 15.6°
61. a C = 40 m/s 2
63. a C = \(\frac{v^{2}}{r} \Rightarrow\) v 2 = r, a C = 78.4, v = 8.85 m/s
T = 5.68 s, which is 0.176 rev/s = 10.6 rev/min
65. Venus is 108.2 million km from the Sun and has an orbital period of 0.6152 y.
r = 1.082 x 10 11 m, T = 1.94 x 10 7 s
v = 3.5 x 10 4 m/s, a C = 1.135 x 10 −2 m/s 2
67. 360 rev/min = 6 rev/s
v = 3.8 m/s, a C = 144. m/s 2
69. a. O′(t) = (4.0 \(\hat{i}\) + 3.0 \(\hat{j}\) + 5.0 \(\hat{k}\))t m
b. \(\vec{r}_{PS}\) = \(\vec{r}_{PS'}\) + \(\vec{r}_{S'S}\), \(\vec{r}\)(t) = \(\vec{r′}\)(t) + (4.0 \(\hat{i}\) + 3.0 \(\hat{j}\) + 5.0 \(\hat{k}\))t m
c. \(\vec{v}\)(t) = \(\vec{v′}\)(t) + (4.0 \(\hat{i}\) + 3.0 \(\hat{j}\) + 5.0 \(\hat{k}\)) m/s
d. The accelerations are the same.
71. \(\vec{v}_{PC}\) = (2.0 \(\hat{i}\) + 5.0 \(\hat{j}\) + 4.0 \(\hat{k}\))m/s
73. a. A = air, S = seagull, G = ground
\(\vec{v}_{SA}\) = 9.0 m/s, velocity of seagull with respect to still air
\(\vec{v}_{AG}\) = ?, \(\vec{v}_{SG}\) = 5 m/s, \(\vec{v}_{SG} = \vec{v}_{SA} + \vec{v}_{AG} \Rightarrow \vec{v}_{AG} = \vec{v}_{SG} − \vec{v}_{SA}\)
\(\vec{v}_{AG}\) = −4.0 m/s
b. \(\vec{v}_{SG} = \vec{v}_{SA} + \vec{v}_{AG} \Rightarrow \vec{v}_{SG}\) = −13.0 m/s
\(\frac{−6000\; m}{−13.0\; m/s}\) = 7 min 42 s
75. Take the positive direction to be the same direction that the river is flowing, which is east. S = shore/Earth, W = water, and B = boat.
a. \(\vec{v}_{BS}\) = 11 km/h, t = 8.2 min
b. \(\vec{v}_{BS}\) = −5 km/h, t = 18 min
c. \(\vec{v}_{BS} = \vec{v}_{BW} + \vec{v}_{WS}, \theta\) = 22° west of north
d. |\(\vec{v}_{BS}\)| = 7.4 km/h, t = 6.5 min
e. \(\vec{v}_{BS}\) = 8.54 km/h, but only the component of the velocity straight across the river is used to get the time
t = 6.0 min
Downstream = 0.3 km
77. \(\vec{v}_{AG} = \vec{v}_{AC} + \vec{v}_{CG}\)
|\(\vec{v}_{AC}\)| = 25 km/h, |\(\vec{v}_{CG}\)| = 15 km/h, |\(\vec{v}_{AG}\)| = 29.15 km/h, \(\vec{v}_{AG} = \vec{v}_{AC} + \vec{v}_{CG}\)
The angle between \(\vec{v}_{AC}\) and \(\vec{v}_{AG}\) is 31°, so the direction of the wind is 14° north of east.
Additional Problems
79. a C = 39.6 m/s 2
81. 90.0 km/h = 25.0 m/s, 9.0 km/h = 2.5 m/s, 60.0 km/h = 16.7 m/s
a T = −2.5 m/s 2 , a C = 1.86 m/s 2 , a = 3.1 m/s 2
83. The radius of the circle of revolution at latitude \(\lambda\) is R E cos \(\lambda\). The velocity of the body is \(\frac{2 \pi r}{T}\). a C = \(\frac{4 \pi^{2} R_{E} \cos \lambda}{T^{2}}\) for \(\lambda\) = 40°, a C = 0.26% g
85. a T = 3.00 m/s 2
v(5 s) = 15.00 m/s, a C = 150.00 m/s 2 , \(\theta\) = 88.8° with respect to the tangent to the circle of revolution directed inward.
|\(\vec{a}\)| = 150.03 m/s 2
87. \(\vec{a}\)(t) = −A\(\omega^{2}\) cos \(\omega\)t \(\hat{i}\) − A\(\omega^{2}\) sin \(\omega\)t \(\hat{j}\)
a C = 5.0 m\(\omega^{2}\), \(\omega\) = 0.89 rad/s
\(\vec{v}\)(t) = −2.24 m/s \(\hat{i}\) − 3.87 m/s \(\hat{j}\)
89. \(\vec{r}_{1}\) = 1.5 \(\hat{j}\) + 4.0 \(\hat{k}\), \(\vec{r}_{2} = \Delta \vec{r} + \vec{r}_{1}\) = 2.5 \(\hat{i}\) + 4.7 \(\hat{j}\) + 2.8 \(\hat{k}\)
91. v x (t) = 265.0 m/s, v y (t) = 20.0 m/s, \(\vec{v}\)(5.0 s) = (265.0 \(\hat{i}\) + 20.0 \(\hat{j}\))m/s
93. R = 1.07 m
95. v 0 = 20.1 m/s
97. v = 3072.5 m/s, a C = 0.223 m/s 2
Challenge Problems
99. a.  a. \(\vec{r}_{TC}\) = (−32 + 80t) \(\hat{i}\) + 50t \(\hat{j}\), |\(\vec{r}_{TC}\)| 2 = (−32 + 80t) 2 + (50t) 2
2r\(\frac{dr}{dt}\) = 2(−32 + 80t) + 100t, \(\frac{dr}{dt} = \frac{2(−32 + 80t) + 100t}{2r}\) = 0
260t = 64 \(\Rightarrow\) t = 15 min
b. |\(\vec{r}_{TC}\)| = 17 km


10.2: Introduction to Benefits and Benefit Trends
What you’ll learn to do: Discuss benefits and benefit trends
Compensation—and employee expectations—doesn’t end with a paycheck. And the technology, generational and competitive market trends that are impacting other aspects of human resource management are evident in the benefits area as well. In fact, HR compliance firm XpertHR identifies “offering the right employee benefits” as one of the “Top 10 HR challenges of 2019.” [1] In this section, we’ll discuss employee benefits and future benefit trends.
Figure: (Unsplash license;
Tuesday Temptation
via
Unsplash
)
- Mucha, Rachel. " A 2020 Vision: Is Your Benefits Package Ready for the Future? " HR Morning. February 13, 2019. Accessed August 22, 2019.


5.6.1: In Real Life- A Case-Controlled Study
In 2015 there was an outbreak of infants born in Brazil with microcephaly, a devastating condition characterized by an abnormally small head size and severe brain damage. This condition is typically rare, which led researchers to hypothesize that it was related in some way to a Zika virus infection in the mothers during pregnancy. There had been an outbreak in northeastern Brazil of the Zika virus, which is mostly transmitted via mosquitos, but can also be transmitted sexually and from mother to fetus ( Zika Virus , 2014). The researchers recruited 91 cases (babies born with microcephaly) and 173 controls (babies without microcephaly, born around the same time), and then investigated their exposure to the Zika virus. What the researchers found was a strong association between a Zika infection during pregnancy and microcephaly: 32 of the 91 cases were Zika positive, and 0 of the 173 controls had contracted the disease. The researchers also noted that Zika tests often produce a false negative, so it is likely that many more of the mothers whose babies were born with microcephaly had indeed contracted the virus during pregnancy (Schneider, 2020, p. 63).


1.6: Matrix Representation of Complex Numbers
In our studies of complex numbers, we noted that multiplication of a complex number by \(e^{iθ}\) rotates that complex number an angle \(θ\) in the complex plane. This leads to the idea that we might be able to represent complex numbers as matrices with \(e^{iθ}\) as the rotation matrix.
Accordingly, we begin by representing \(e^{iθ}\) as the rotation matrix, that is,
\[\begin{aligned}e^{i\theta}&=\left(\begin{array}{rr}\cos\theta&-\sin\theta \\ \sin\theta&\cos\theta\end{array}\right) \\ &=\cos\theta\left(\begin{array}{cc}1&0\\0&1\end{array}\right)+\sin\theta\left(\begin{array}{rr}0&-1\\1&0\end{array}\right).\end{aligned} \nonumber \]
Since \(e^{iθ} = \cos θ + i \sin θ\), we are led to the matrix representations of the unit numbers as
\[1=\left(\begin{array}{cc}1&0\\0&1\end{array}\right),\quad i=\left(\begin{array}{rr}0&-1\\1&0\end{array}\right).\nonumber \]
A general complex number \(z = x + iy\) is then represented as
\[z=\left(\begin{array}{rr}x&-y\\y&x\end{array}\right).\nonumber \]
The complex conjugate operation, where \(i → −i\), is seen to be just the matrix transpose.
Show that \(i^2=-1\) in the matrix representation.
Solution
We have
\[i^2=\left(\begin{array}{rr}0&-1\\1&0\end{array}\right)\left(\begin{array}{rr}0&-1\\1&0\end{array}\right)=\left(\begin{array}{rr}-1&0\\0&-1\end{array}\right)=-\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=-1.\nonumber \]
Show that \(z\overline{z}=x^2+y^2\) in the matrix representation.
Solution
We have
\[z\overline{z}=\left(\begin{array}{rr}x&-y\\y&x\end{array}\right)\left(\begin{array}{rr}x&y\\-y&x\end{array}\right)=\left(\begin{array}{cc}x^2+y^2&0\\0&x^2+y^2\end{array}\right)=(x^2+y^2)\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=(x^2+y^2).\nonumber \]
We can now see that there is a one-to-one correspondence between the set of complex numbers and the set of all two-by-two matrices with equal diagonal elements and opposite signed off-diagonal elements. If you do not like the idea of \(\sqrt{-1}\), then just imagine the arithmetic of these two-by-two matrices!


18.1: Introduction
The gastrointestinal system is vital to the overall health of the body. The ability of this system to function optimally depends not only on the absence of illness but on the proper development of this system throughout the lifespan. This chapter will examine the nutritional requirements needed at each stage of life to assist this system in developing to its optimal capacity. It will also review illnesses that impact this system.
Consider this case: The client—unborn at the start of this chapter's case study—is Bao Nhung. The client’s mother, Hien Nhung, is a 23-year-old first-generation Vietnamese American who works as a medical–surgical nurse on the night shift at the local hospital. The client’s mother works long nights, and the pregnancy was unplanned. Hien has not received any prenatal care during the pregnancy.


2.3: Green's Function Analysis of Scattering Resonances
Quasi-bound states and resonances are not limited to 1D, but are equally (or more) important in 2D and 3D. A useful and general way to study them is the quantum Green’s function formalism developed in the previous chapter.
Let \(\hat{H} = \hat{T} + \hat{V}\) be the Hamiltonian of a system supporting resonances, where \(\hat{T}\) is the kinetic energy operator and \(\hat{V}\) is the potential operator. We decompose the potential into
\[\hat{V} = \hat{V}_0 + \hat{V}_1,\]
where \(\hat{V}_0\) is a “confining potential” that supports a bound state, and \(\hat{V}_1\) is a “deconfining potential” that turns the bound state into a quasi-bound state. The figure below shows an example of such a decomposition, for the 1D model from the previous section.
For the potential \(\hat{V}_0\) (in the absence of \(\hat{V}_1\) ), let there be a bound state \(|\varphi\rangle\) , with energy \(E_0\) . Furthermore, we assume that the potential supports a continuum of free states \(\{|\psi_k\rangle\}\) with energies \(\{E_k\}\) , where \(k\) is some \(d\) -dimensional continuous index for the free states (analogous to the wave-vector for plane wave states). The bound state and free states satisfy the Schrödinger equation
\[\begin{align} \big(\hat{T} + \hat{V}_0\big) |\varphi\rangle \; &= E_0 |\varphi\rangle \\ \big(\hat{T} + \hat{V}_0\big) |\psi_k\rangle &= E_k |\psi_k\rangle,\end{align}\]
along with the orthogonality and completeness relations
\[\langle\varphi|\psi_k\rangle = 0, \quad |\varphi\rangle\langle\varphi| + \int d^dk\, |\psi_k\rangle\langle\psi_k| = \hat{I}.\]
As described in Section 1.7, the causal Green’s function is
\[\hat{G}_0(E) = \lim_{\varepsilon\rightarrow0^+} \Big(E - \hat{T} - \hat{V}_0 + i\varepsilon\Big)^{-1}.\]
Now introduce the deconfining potential \(\hat{V}_1\) . According to Dyson’s equations (Section VI of the previous chapter), the Green’s function for the full system is
\[\hat{G} = \hat{G}_0 + \hat{G} \hat{V}_1 \hat{G}_0.\]
We want the matrix elements of \(\hat{G}(E)\) , which can then be used to find scattering amplitudes. We will calculate them by using the states \(|\varphi\rangle\) and \(\{|\psi_k\rangle\}\) as a basis—but note that this basis does not consist of energy eigenfunctions of the full Hamiltonian \(\hat{H}\) . In particular, \(|\varphi\rangle\) is not an exact eigenstate of \(\hat{H}\) .
As usual when dealing with Dyson’s equations, we must watch out for the fact that \(\hat{G}\) appears on both the left and right hand sides. This can be dealt with by judiciously inserting a resolution of the identity:
\[\begin{align} \begin{aligned}\hat{G} &= \hat{G}_0 + \hat{G} \hat{I} \hat{V}_1 \hat{G}_0 \\ &= \hat{G}_0 + \hat{G} \left(|\varphi\rangle\langle\varphi| + \int d^dk\, |\psi_k\rangle\langle\psi_k|\right) \hat{V}_1 \hat{G}_0. \end{aligned}\end{align}\]
We now compute the matrix element \(\langle\varphi|\cdots|\varphi\rangle\) for both sides of the equation:
\[\begin{align} \nonumber \langle\varphi|\hat{G}|\varphi\rangle = \langle\varphi|\hat{G}_0|\varphi\rangle &+ \langle\varphi|\hat{G}|\varphi\rangle \, \langle\varphi|\hat{V}_1 \hat{G}_0|\varphi\rangle + \int d^dk\, \langle\varphi|\hat{G}|\psi_k\rangle \, \langle\psi_k| \hat{V}_1 \hat{G}_0|\varphi\rangle \nonumber\\ \nonumber \langle\varphi|\hat{G}|\varphi\rangle \left(1 - \langle\varphi|\hat{V}_1 \hat{G}_0|\varphi\rangle\right) &= \langle\varphi|\hat{G}_0|\varphi\rangle + \int d^dk\, \langle\varphi|\hat{G}|\psi_k\rangle \, \langle\psi_k| \hat{V}_1 \hat{G}_0|\varphi\rangle \nonumber\\ \nonumber \lim_{\varepsilon\rightarrow0^+} \langle\varphi|\hat{G}|\varphi\rangle \left(1 - \frac{\langle\varphi|\hat{V}_1|\varphi\rangle}{E - E_0 + i\varepsilon}\right) &= \lim_{\varepsilon\rightarrow0^+} \frac{1}{E - E_0 + i\varepsilon} \left(1+ \int d^dk\, \langle\varphi|\hat{G}|\psi_k\rangle \, \langle\psi_k| \hat{V}_1|\varphi\rangle \right) \nonumber\\ \lim_{\varepsilon\rightarrow0^+} \langle\varphi|\hat{G}|\varphi\rangle \Big(E - E_0 -\, \langle\varphi|\hat{V}_1|\varphi\rangle \, & + i\varepsilon\Big) - \int d^dk\, \langle\varphi|\hat{G}|\psi_k\rangle \, \langle\psi_k| \hat{V}_1|\varphi\rangle = 1. \label{psiGphi}\end{align}\]
Similarly, computing the matrix element \(\langle\varphi|\cdots|\psi_k\rangle\) gives:
\[\begin{align} \begin{aligned} \langle\varphi|\hat{G}|\psi_k\rangle &= \langle\varphi|\hat{G}_0|\psi_k\rangle + \langle\varphi|\hat{G}|\varphi\rangle \, \langle\varphi|\hat{V}_1 \hat{G}_0|\psi_k\rangle + \int d^dk'\, \langle\varphi|\hat{G}|\psi_{k'}\rangle \, \langle\psi_{k'}| \hat{V}_1 \hat{G}_0|\psi_k\rangle \\ &= \lim_{\varepsilon\rightarrow0^+} \left(E-E_k+i\varepsilon\right)^{-1} \left(\langle\varphi|\hat{G}|\varphi\rangle \, \langle\varphi|\hat{V}_1|\psi_k\rangle + \int d^dk'\, \langle\varphi|\hat{G}|\psi_{k'}\rangle \, \langle\psi_{k'}| \hat{V}_1|\psi_k\rangle\right).\end{aligned}\end{align}\]
The equations thus far have been exact (we have not used perturbation theory). Now we apply an approximation: in the last line of the above equation, let the factor of \(\langle\varphi|G|\varphi\rangle\) be large, so that the first term in the sum becomes dominant. It will be shown below that \(\langle\varphi|G|\varphi\rangle\) being large is precisely the resonance condition, so this approximation will be self-consistent. With this, we obtain
\[\langle\varphi|\hat{G}|\psi_k\rangle \approx \lim_{\varepsilon\rightarrow0^+} \frac{\langle\varphi|\hat{G}|\varphi\rangle \, \langle\varphi|\hat{V}_1|\psi_k\rangle}{E-E_k+i\varepsilon}.\]
Combining this with Equation \(\eqref{psiGphi}\) gives
\[\lim_{\varepsilon\rightarrow0^+} \left[\langle\varphi|\hat{G}|\varphi\rangle \left(E - E_0 -\, \langle\varphi|\hat{V}_1|\varphi\rangle \, + i\varepsilon\right) - \int d^dk\, \frac{\langle\varphi|\hat{G}|\varphi\rangle\langle\varphi|\hat{V}_1|\psi_k\rangle}{E-E_k+i\varepsilon} \, \langle\psi_k| \hat{V}_1|\varphi\rangle\right] \approx 1.\]
Hence,
Definition: Resonance Condition
\[\begin{align}\begin{aligned} \langle\varphi|\,\hat{G}(E)\,|\varphi\rangle \approx \frac{1}{\displaystyle E - E_0 - \langle\varphi|V_1|\varphi\rangle - \Sigma(E)} \\ \mathrm{where}\;\;\Sigma(E) \equiv \lim_{\varepsilon\rightarrow0^+} \int d^dk\, \frac{\displaystyle| \langle\psi_k| \hat{V}_1|\varphi\rangle|^2}{\displaystyle E-E_k+i\varepsilon}. \end{aligned} \label{phiGphi_result}\end{align}\]
The quantity \(\Sigma(E)\) is called the self-energy , and we will have more to say about it shortly. It depends on \(E\) , but let us assume for now that the dependence is weak, so that \(\Sigma\) can be effectively treated as a constant. It is complex-valued, and both its real and imaginary parts are important; we will shortly show that \(\mathrm{Im}(\Sigma) < 0\) .
From Equation \(\eqref{phiGphi_result}\), we can see that \(\langle\varphi|\hat{G}(E)|\varphi\rangle\) is large when the denominator is as close to zero as possible. This is called the resonance condition , and is self-consistent with the approximation that we made in the above derivation. As we vary the incident energy \(E\) over the range of real values, the resonance condition is satisfied when
\[E \;\approx\; E_{\mathrm{res}} \,\equiv\, E_0 + \langle\varphi|\hat{V}_1|\varphi\rangle + \mathrm{Re}\big[\,\Sigma\,\big].\]
We call \(E_{\mathrm{res}}\) the resonance energy . Its first term is the energy of the original bound state, in the absence of the deconfining potential \(\hat{V}_1\) . The second term is the energy shift induced by \(\hat{V}_1\) . The third term is equal to the real part of the self-energy \(\Sigma\) , and has a more subtle meaning. Since the definition of \(\Sigma\) involves \(\{|\psi_k\rangle\}\) , we can think of this term as an energy shift induced by the continuum of free states.
In Section 1.8, we derived the following relationship between the Green’s function and the scattering amplitude \(f\) :
\[f(\mathbf{k}\rightarrow\mathbf{k}') \;\propto\; \langle \mathbf{k}'|\hat{V}|\mathbf{k}\rangle + \langle \mathbf{k}'|\hat{V}\hat{G}\hat{V}|\mathbf{k}\rangle.\]
Here, \(|\mathbf{k}\rangle\) and \(|\mathbf{k}'\rangle\) are incident and scattered plane-wave states satisfying \(|\mathbf{k}|=|\mathbf{k}'|\) . The first term describes the lowest-order scattering process (the first Born approximation). The second term contains all second- and higher-order scattering processes. By inserting resolutions of the identity between each \(\hat{V}\) and \(\hat{G}\) operator in the second term, we find that \(f\) contains a contribution of the form
\[\Delta f(\mathbf{k}\rightarrow\mathbf{k}') \;\propto\; \langle \mathbf{k}'|\hat{V}|\varphi\rangle\langle\varphi|\hat{G}|\varphi\rangle\langle\varphi|\hat{V}|\mathbf{k}\rangle \;=\; \frac{\langle \mathbf{k}'|\hat{V}|\varphi\rangle \, \langle\varphi|\hat{V}|\mathbf{k}\rangle}{\displaystyle E - E_{\mathrm{res}} - i \mathrm{Im}[\,\Sigma\,]}. \label{deltaf}\]
At resonance, the denominator becomes small and hence \(\Delta f\) should be the dominant contribution to \(f\) . It is worth emphasizing that \(\Delta f\) is extracted from a combination of all terms in the Born series, not just low-order terms. Intuitively, we can think of a resonant scattering scenario as one where the particle bounces around inside the potential many times before it finally escapes—i.e., high orders in the Born series are significant.
The figure below shows the energy dependence of \(\Delta f\) , according to Equation \(\eqref{deltaf}\):
The graph of \(|\Delta f|^2\) versus \(E\) has a shape known as a Lorentzian . It has a peak centered at the resonance energy \(E_{\mathrm{res}}\) . The peak width is characterized by the full-width at half-maximum (FWHM), the spacing between the two energies where \(|\Delta f|^2\) is at half its maximum value:
\[\delta E^{(\mathrm{FWHM})} = 2\, \Big|\mathrm{Im}[\Sigma]\Big|.\]
Thus, the closer the self-energy gets to being a real quantity, the sharper the peak.
The phase \(\mathrm{arg}[\Delta f]\) also contains useful information. As \(E\) crosses \(E_{\mathrm{res}}\) from below, the phase increases by \(\pi\) . The energy range over which this phase shift occurs is \(\sim |\mathrm{Im}[\Sigma]|\) .
These two signatures—peaks and phase shifts—are sought after in numerous real-world scattering experiments. In actual experiments, the peaks and phase shifts are often overlaid on a “background” caused by non-resonant effects. For example, the plot below was released by the CMS experiment at the Large Hadron Collider (LHC), showing a resonance peak on a large background. This was part of the evidence for the LHC’s discovery of a new particle, the Higgs boson, in 2012.


Problem Set 1: Use the Point Formula for Demand Elasticities.
Exercise \(\PageIndex{1}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 150 -2P_{1} .5 P_{2} + 0.1M\)
\(P_{1} = 20\)
\(P_{2} = 30\)
\(M = 100\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -0.53\)
\(\varepsilon_{12} = -0.6\)
\(\varepsilon_{1M} = 0.13\)
Inelastic
Complement
Normal necessity
Exercise \(\PageIndex{2}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 75-2.5P_{1} - 0.5 P_{2} + 0.5M\)
\(P_{1} = 30\)
\(P_{2} = 40\)
\(M = 200\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -0.94\)
\(\varepsilon_{12} = -0.25\)
\(\varepsilon_{1M} = 1.25\)
Inelastic
Complement
Normal luxury
Exercise \(\PageIndex{3}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 150-3P_{1} + 0.5 P_{2} - 0.5M\)
\(P_{1} = 40\)
\(P_{2} = 50\)
\(M = 300\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -4.8\)
\(\varepsilon_{12} = 1\)
\(\varepsilon_{1M} = -1.2\)
Elastic
Substitute
Inferior
Exercise \(\PageIndex{4}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 100-2P_{1} + 1.5 P_{2} - 0.2M\)
\(P_{1} = 50\)
\(P_{2} = 40\)
\(M = 200\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -5\)
\(\varepsilon_{12} = 3\)
\(\varepsilon_{1M} = -2\)
Elastic
Substitute
Inferior
Exercise \(\PageIndex{5}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 90-2.5P_{1} -0.5 P_{2} +0.5 M\)
\(P_{1} = 30\)
\(P_{2} = 40\)
\(M = 200\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -0.79\)
\(\varepsilon_{12} = -0.21\)
\(\varepsilon_{1M} = 1.05\)
Inelastic
Complement
Normal luxury
Exercise \(\PageIndex{6}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 170-2.5P_{1} -1.5 P_{2} +0.1 M\)
\(P_{1} = 40\)
\(P_{2} = 30\)
\(M = 100\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -2.86\)
\(\varepsilon_{12} = -1.29\)
\(\varepsilon_{1M} = 0.29\)
Elastic
Complement
Normal necessity
Exercise \(\PageIndex{7}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 100-3P_{1} -0.5 P_{2} +0.5 M\)
\(P_{1} = 40\)
\(P_{2} = 20\)
\(M = 300\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
Answers:
\(\varepsilon_{11} = -1\)
\(\varepsilon_{12} = -0.08\)
\(\varepsilon_{1M} = 1.25\)
Unitary elastic
Complement
Normal luxury
Exercise \(\PageIndex{8}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 200-2P_{1} +0.5 P_{2} -0.1 M\)
\(P_{1} = 20\)
\(P_{2} = 10\)
\(M = 200\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -0.28\)
\(\varepsilon_{12} = -0.03\)
\(\varepsilon_{1M} = -0.14\)
Inelastic
Substitute
Inferior
Exercise \(\PageIndex{9}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 200-2.5P_{1} +1.5 P_{2} -0.5 M\)
\(P_{1} = 10\)
\(P_{2} = 20\)
\(M = 100\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -0.16\)
\(\varepsilon_{12} = -0.19\)
\(\varepsilon_{1M} = -0.32\)
Inelastic
Substitute
Inferior
Exercise \(\PageIndex{10}\)
Given the following:
\(\textrm{The demand equation is:} Q_{1}= 60-3P_{1} +1P_{2} +0.1 M\)
\(P_{1} = 20\)
\(P_{2} = 30\)
\(M = 300\)
\(\textrm{Calculate} \varepsilon _{11}, \varepsilon_{12}, \varepsilon_{1M}\)
Is demand for Good 1 inelastic, unitary elastic, or elastic?
Is Good 2 a substitute or complement to good 1?
Is Good 1 inferior, a normal necessity, or a normal luxury?
-
Answer
-
\(\varepsilon_{11} = -1\)
\(\varepsilon_{12} = -0.5\)
\(\varepsilon_{1M} = -0.5\)
Unitary elastic
Substitute
Normal necessity
Problem Set 2: Use Elasticities to Determine Whether to Increase Price.
Exercise \(\PageIndex{1}\)
Given the following:
Your current quantity is 1000 units.
Your current price is 25 dollars.
Your average variable cost is 15 dollars per unit.

What will happen to profit above variable cost if you raise your price by 2 dollars?
-
Answer
-
Your current profit above variable cost is 10000 dollars.
Your price will change by 8 percent.
Your new quantity is 760 units.
If you raise price, the profit above variable cost will be 9120 dollars.
Exercise \(\PageIndex{2}\)
Given the following:
Your current quantity is 1100 units.
Your current price is 40 dollars.
Your average variable cost is 36 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 4400 dollars.
Your price will change by 2.5 percent.
Your new quantity is 990 units.
If you raise price, the profit above variable cost will be 4950 dollars.
Exercise \(\PageIndex{3}\)
Given the following:
Your current quantity is 1120 units.
Your current price is 50 dollars.
Your average variable cost is 40 dollars per unit.

What will happen to profit above variable cost if you raise your price by 2 dollars?
-
Answer
-
Your current profit above variable cost is 11200 dollars.
Your price will change by 4 percent.
Your new quantity is 896 units.
If you raise price, the profit above variable cost will be 10752 dollars.
Exercise \(\PageIndex{4}\)
Given the following:
Your current quantity is 1200 units.
Your current price is 20 dollars.
Your average variable cost is 16 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 4800 dollars.
Your price will change by 5 percent.
Your new quantity is 960 units.
If you raise price, the profit above variable cost will be 4800 dollars.
Exercise \(\PageIndex{5}\)
Given the following:
Your current quantity is 1300 units.
Your current price is 50 dollars.
Your average variable cost is 40 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 13000 dollars.
Your price will change by 2 percent.
Your new quantity is 1261 units.
If you raise price, the profit above variable cost will be 13871 dollars.
Exercise \(\PageIndex{6}\)
Given the following:
Your current quantity is 1200 units.
Your current price is 20 dollars.
Your average variable cost is 16 dollars per unit.

What will happen to profit above variable cost if you raise your price by 2 dollars?
-
Answer
-
Your current profit above variable cost is 4800 dollars.
Your price will change by 10 percent.
Your new quantity is 1110 units.
If you raise price, the profit above variable cost will be 6660 dollars.
Exercise \(\PageIndex{7}\)
Given the following:
Your current quantity is 1120 units.
Your current price is 50 dollars.
Your average variable cost is 46 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 4480 dollars.
Your price will change by 2 percent.
Your new quantity is 1097.6 units.
If you raise price, the profit above variable cost will be 5488 dollars.
Exercise \(\PageIndex{8}\)
Given the following:
Your current quantity is 1200 units.
Your current price is 25 dollars.
Your average variable cost is 21 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 4800 dollars.
Your price will change by 4 percent.
Your new quantity is 960 units.
If you raise price, the profit above variable cost will be 4800 dollars.
Exercise \(\PageIndex{9}\)
Given the following:
Your current quantity is 1050 units.
Your current price is 50 dollars.
Your average variable cost is 40 dollars per unit.

What will happen to profit above variable cost if you raise your price by 2 dollars?
-
Answer
-
Your current profit above variable cost is 10500 dollars.
Your price will change by 4 percent.
Your new quantity is 819 units.
If you raise price, the profit above variable cost will be 9828 dollars.
Exercise \(\PageIndex{10}\)
Given the following:
Your current quantity is 1000 units.
Your current price is 25 dollars.
Your average variable cost is 21 dollars per unit.

What will happen to profit above variable cost if you raise your price by 1 dollars?
-
Answer
-
Your current profit above variable cost is 4000 dollars.
Your price will change by 4 percent.
Your new quantity is 820 units.
If you raise price, the profit above variable cost will be 4100 dollars.
Problem Set 3: Use Elasticities to Determine Whether to Increase Advertising.
Exercise \(\PageIndex{1}\)
Given the following:
Your current quantity is 1000 units.
Your price is 25 dollars.
Your average variable cost (AVC) is 15 dollars per unit.
Your elasticity of demand with respect to advertising is 0.4.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 200 dollars?
-
Answer
-
Your current profit is 5000 dollars.
Your advertising will change by 4 percent.
Your new quantity is 1016 units.
Profit will go down to 4960 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{2}\)
Given the following:
Your current quantity is 2000 units.
Your price is 40 dollars.
Your average variable cost (AVC) is 36 dollars per unit.
Your elasticity of demand with respect to advertising is 0.5.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 300 dollars?
-
Answer
-
Your current profit is 3000 dollars.
Your advertising will change by 6 percent.
Your new quantity is 2060 units.
Profit will go down to 2940 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{3}\)
Given the following:
Your current quantity is 1500 units.
Your price is 50 dollars.
Your average variable cost (AVC) is 40 dollars per unit.
Your elasticity of demand with respect to advertising is 0.6.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 400 dollars?
-
Answer
-
Your current profit is 10000 dollars.
Your advertising will change by 8 percent.
Your new quantity is 1572 units.
Profit will go up to 10320 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{4}\)
Given the following:
Your current quantity is 1200 units.
Your price is 20 dollars.
Your average variable cost (AVC) is 16 dollars per unit.
Your elasticity of demand with respect to advertising is 0.5.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 250 dollars?
-
Answer
-
Your current profit is -200 dollars.
Your advertising will change by 5 percent.
Your new quantity is 1230 units.
Profit will go down to -330 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{5}\)
Given the following:
Your current quantity is 1300 units.
Your price is 50 dollars.
Your average variable cost (AVC) is 40 dollars per unit.
Your elasticity of demand with respect to advertising is 1.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 200 dollars?
-
Answer
-
Your current profit is 8000 dollars.
Your advertising will change by 4 percent.
Your new quantity is 1352 units.
Profit will go up to 8320 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{6}\)
Given the following:
Your current quantity is 2000 units.
Your price is 20 dollars.
Your average variable cost (AVC) is 16 dollars per unit.
Your elasticity of demand with respect to advertising is 1.5.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 300 dollars?
-
Answer
-
Your current profit is 3000 dollars.
Your advertising will change by 6 percent.
Your new quantity is 2180 units.
Profit will go up to 3420 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{7}\)
Given the following:
Your current quantity is 1500 units.
Your price is 50 dollars.
Your average variable cost (AVC) is 46 dollars per unit.
Your elasticity of demand with respect to advertising is 0.4.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 400 dollars?
-
Answer
-
Your current profit is 1000 dollars.
Your advertising will change by 8 percent.
Your new quantity is 1548 units.
Profit will go down to 792 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{8}\)
Given the following:
Your current quantity is 1200 units.
Your price is 25 dollars.
Your average variable cost (AVC) is 21 dollars per unit.
Your elasticity of demand with respect to advertising is 0.5.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 250 dollars?
-
Answer
-
Your current profit is -200 dollars.
Your advertising will change by 5 percent.
Your new quantity is 1230 units.
Profit will go down to -330 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{9}\)
Given the following:
Your current quantity is 2000 units.
Your price is 50 dollars.
Your average variable cost (AVC) is 40 dollars per unit.
Your elasticity of demand with respect to advertising is 2.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 200 dollars?
-
Answer
-
Your current profit is 15000 dollars.
Your advertising will change by 4 percent.
Your new quantity is 2160 units.
Profit will go up to 16400 dollars (after subtracting the increse in advertising).
Exercise \(\PageIndex{10}\)
Given the following:
Your current quantity is 1000 units.
Your price is 25 dollars.
Your average variable cost (AVC) is 21 dollars per unit.
Your elasticity of demand with respect to advertising is 1.5.
Your current advertising expenditure is $5,000 and is your only fixed cost.
What will happen to profit if you increase advertising by 300 dollars?
-
Answer
-
Your current profit is -1000 dollars.
Your advertising will change by 6 percent.
Your new quantity is 1090 units.
Profit will go up to -940 dollars (after subtracting the increse in advertising).
Problem Set 4: Short-Run and Long-Run Elasticities.
Exercise \(\PageIndex{1}\)
Given the following:
The short run demand equation is:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 2P_{1} + 0.5 Q_{1}^{(t-1)}\)
\(P_{1} = 35\)
\(Q_{1}^{(t-1)} =60\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.5\)
\(Q_{1}^{LR} = 200-4P_{1}\)
\(\varepsilon_{11}^{SR} = -1.17\)
\(\varepsilon_{11}^{LR} = -2.33\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{2}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 3P_{1} + 0.6 Q_{1}^{(t-1)}\)
\(P_{1} = 20\)
\(Q_{1}^{(t-1)} =100\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.6\)
\(Q_{1}^{LR} = 250-7.5P_{1}\)
\(\varepsilon_{11}^{SR} = -0.6\)
\(\varepsilon_{11}^{LR} = -1.5\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{3}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 150 - 4P_{1} + 0.75 Q_{1}^{(t-1)}\)
\(P_{1} = 20\)
\(Q_{1}^{(t-1)} =280\)
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.75\)
\(Q_{1}^{LR} = 600-16P_{1}\)
\(\varepsilon_{11}^{SR} = -0.29\)
\(\varepsilon_{11}^{LR} = -1.14\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{4}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 250 - 4P_{1} + 0.8Q_{1}^{(t-1)}\)
\(P_{1} = 40\)
\(Q_{1}^{(t-1)} =450\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.8\)
\(Q_{1}^{LR} = 1250-20P_{1}\)
\(\varepsilon_{11}^{SR} = -0.36\)
\(\varepsilon_{11}^{LR} = -1.78\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{5}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 120 - 2P_{1} + 0.6Q_{1}^{(t-1)}\)
\(P_{1} = 40\)
\(Q_{1}^{(t-1)} =100\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.6\)
\(Q_{1}^{LR} = 300-5P_{1}\)
\(\varepsilon_{11}^{SR} = -0.8\)
\(\varepsilon_{11}^{LR} = -2\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{6}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 150 - 3P_{1} + 0.8Q_{1}^{(t-1)}\)
\(P_{1} = 30\)
\(Q_{1}^{(t-1)} =300\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.8\)
\(Q_{1}^{LR} = 750-15P_{1}\)
\(\varepsilon_{11}^{SR} = -0.3\)
\(\varepsilon_{11}^{LR} = -1.5\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{7}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 200 - 3P_{1} + 0.75Q_{1}^{(t-1)}\)
\(P_{1} = 45\)
\(Q_{1}^{(t-1)} =260\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.75\)
\(Q_{1}^{LR} = 800-12P_{1}\)
\(\varepsilon_{11}^{SR} = -0.52\)
\(\varepsilon_{11}^{LR} = -2.08\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{8}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 80 - 2P_{1} + 0.6Q_{1}^{(t-1)}\)
\(P_{1} = 30\)
\(Q_{1}^{(t-1)} =50\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.6\)
\(Q_{1}^{LR} = 200-5P_{1}\)
\(\varepsilon_{11}^{SR} = -1.2\)
\(\varepsilon_{11}^{LR} = -3\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{9}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 120 - 3P_{1} + 0.5Q_{1}^{(t-1)}\)
\(P_{1} = 30\)
\(Q_{1}^{(t-1)} =60\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.5\)
\(Q_{1}^{LR} = 240-6P_{1}\)
\(\varepsilon_{11}^{SR} = -1.5\)
\(\varepsilon_{11}^{LR} = -3\)
\(\textrm{In the long-run.}\)
Exercise \(\PageIndex{10}\)
Given the following:
\(\textrm{The short run demand equation is:} Q_{1}^{(t)} = 100 - 2P_{1} + 0.5Q_{1}^{(t-1)}\)
\(P_{1} = 40\)
\(Q_{1}^{(t-1)} =40\)
What is the habit parameter?
What is the long-run demand equation?
What is the short-run own-price elasticity of demand?
What is the long-run own-price elasticity of demand?
When is demand more elastic, in the short- or long-run?
-
Answer
-
\(\textrm{Habit parameter}: \gamma = 0.5\)
\(Q_{1}^{LR} = 200-4P_{1}\)
\(\varepsilon_{11}^{SR} = -2\)
\(\varepsilon_{11}^{LR} = -4\)
\(\textrm{In the long-run.}\)
Problem Set 5: Multiple Choice.
Exercise \(\PageIndex{1}\)
-
When is demand unitary elastic?
a) When the income elasticity is 1.
b) When the own-price elasticity of demand is -1.
c) When demand is totally unresponsive (inverse demand is a vertical line).
d) All of the above.
-
Answer
-
b
Exercise \(\PageIndex{2}\)
-
If price goes up by 4 percent and quantity demanded falls by 2 percent then
a) This is a substitute good.
b) This good has an elastic demand.
c) This good is an inferior good.
d) This good has an inelastic demand.
-
Answer
-
d
Exercise \(\PageIndex{3}\)
-
Suppose that the own-price elasticity of demand is -1. Which statement is true?
a) Profits are maximized.
b) Revenue is maximized.
c) Consumer surplus is maximized.
d) All of the above.
-
Answer
-
b
Exercise \(\PageIndex{4}\)
-
Suppose the own-price elasticity of demand is -0.75. Which statement is true?
a) An increase in price will decrease revenue.
b) An increase in price will increase revenue.
c) The good in question is a substitute in consumption.
d) The good in question is an inferior good.
-
Answer
-
b
Exercise \(\PageIndex{5}\)
-
Which of the following elasticity numbers is consistent with a normal good?
a) An income elasticity of demand that is -0.5.
b) An income elasticity of demand that is 0.5.
c) An own-price elasticity of 0.28.
d) Both a and c.
-
Answer
-
b
Exercise \(\PageIndex{6}\)
-
If an increase in income causes an increase in demand, then we know that
a) The good in question is a normal luxury good.
b) The good in question is a normal necessity good.
c) The good in question is a normal good but we can’t tell whether it is classified as a necessity or a luxury without further information.
d) The good in question violates the law of demand.
-
Answer
-
c
Exercise \(\PageIndex{7}\)
-
Which demand elasticity number tells you that two products are substitutes in consumption?
a) A cross-price elasticity of -0.5.
b) A cross-price elasticity of 0.5.
c) An income elasticity of 1.5.
d) An own-price elasticity of -0.75.
-
Answer
-
b
Exercise \(\PageIndex{8}\)
-
One problem with using the arc elasticity formula to compute an own-price elasticity is:
a) It is very hard to compute because you need lots of data points.
b) Actually, there are no problems with using the arc formula to compute elasticities.
c) One must assume that all other factors that affect demand, other than own price, remain the same.
d) If demand has not shifted, the arc formula will often return a positive value for the own-price elasticity.
-
Answer
-
c
Exercise \(\PageIndex{9}\)
-
Given that a firm has some control over the price it charges (it is not a price taker) and it faces a positive marginal cost, which case would be consistent with profit maximization?
a) It sets its price to make demand inelastic.
b) It sets its price to make demand elastic.
c) It sets its price to make demand unitary elastic.
d) It sets its price so that both demand and supply are in the inelastic range.
-
Answer
-
b
Exercise \(\PageIndex{10}\)
-
As described in class, the supply schedule for van Gogh paintings:
a) Is perfectly inelastic.
b) Is downward sloping.
c) Maybe quite inelastic but is probably not perfectly inelastic.
d) Is a horizontal line.
e) Choices a and d only.
-
Answer
-
c
Exercise \(\PageIndex{11}\)
-
Other things equal, in the long run
a) Demand is less elastic than in the short run.
b) Supply is less elastic than in the short run.
c) Demand is more elastic than in the short run.
d) Both demand and supply are less elastic than in the short run.
-
Answer
-
c
Exercise \(\PageIndex{12}\)
-
Which best describes the concept of elasticity?
a) The responsiveness of demand or supply to own-price or some other shift variable.
b) The idea that demand schedules always slope downwards.
c) The idea that consumer surplus is the area under the demand schedule.
d) The idea that supply schedules always have non-negative slopes.
-
Answer
-
a


1.5: Energy
- Define heat and work.
- Distinguish between kinetic energy and potential energy.
- State the law of conservation of matter and energy.
Just like matter, energy is a term that we are all familiar with and use on a daily basis.  When you plug a lamp into an electric socket, you see energy in the form of light, but when you plug a heating pad into that same socket, you only feel warmth. Without energy, we could not turn on lights, make our lunch, nor travel to school. In fact, without energy, we could not even wake up because our bodies require energy to function. We use energy for every single thing that we do, whether we are awake or asleep.
Ability to Do Work or Produce Heat
When we speak of using energy, we are really referring to transferring energy from one place to another. When you use energy to throw a ball, you transfer energy from your body to the ball, and this causes the ball to fly through the air. When you use energy to warm your house, you transfer energy from the furnace to the air in your home, and this causes the temperature in your house to rise. Although energy is used in many kinds of different situations, all of these uses rely on energy being transferred in one of two ways. Energy can be transferred as heat or as work .
When scientists speak of heat , they are referring to energy that is transferred from an object with a higher temperature to an object with a lower temperature as a result of the temperature difference. Heat will "flow" from the hot object to the cold object until both end up at the same temperature. When you cook with a metal pot, you witness energy being transferred in the form of heat. Initially, only the stove element is hot—the pot and the food inside the pot are cold. As a result, heat moves from the hot stove element to the cold pot. After a while, enough heat is transferred from the stove to the pot, raising the temperature of the pot and all of its contents (Figure \(\PageIndex{1}\)).
Heat is one way in which energy can be transferred. Energy can also be transferred as work . The scientific definition of work is force (any push or pull) applied over a distance . Whenever you push an object and cause it to move, you have done work, and transferred some energy to the object. So far, we've talked about the two ways in which energy can be transferred from one place, or object, to another. Energy can be transferred as heat, and energy can be transferred as work.
Kinetic Energy
Machines use energy, our bodies use energy, energy comes from the sun, energy comes from volcanoes, energy causes forest fires, and energy helps us to grow food. With all these seemingly different types of energy, it's hard to believe that there are really only two different forms of energy: kinetic energy and potential energy . Kinetic energy is energy associated with motion. When an object is moving, it has kinetic energy. When the object stops moving, it has no kinetic energy. While all moving objects have kinetic energy, not all moving objects have the same amount of kinetic energy. The amount of kinetic energy possessed by an object is determined by its mass and its speed. The heavier an object is and the faster it is moving, the more kinetic energy it has.
Kinetic energy is very common, and it's easy to spot examples of it in the world around you. Sometimes we even try to capture kinetic energy and use it to power things like our home appliances. If you are from California, you might have driven through the Tehachapi Pass near Mojave or the Montezuma Hills in Solano County and seen the windmills lining the slopes of the mountains (Figure \(\PageIndex{2}\)). These are two of the larger wind farms in North America. As wind rushes along the hills, the kinetic energy of the moving air particles turns the windmills, trapping the wind's kinetic energy so that people can use it in their houses and offices.
Potential Energy
Potential energy is stored energy. It is energy that remains available until we choose to use it. Think of a battery in a flashlight. If you leave a flashlight on, the battery will run out of energy within a couple of hours, and your flashlight will die. If, however, you only use the flashlight when you need it, and you turn it off when you don’t, the battery will last for days or even months. The battery contains a certain amount of energy, and it will power the flashlight for a certain amount of time, but because the battery stores potential energy, you can choose to use the energy all at once, or you can save it and only use a small amount at a time.
Any stored energy is potential energy. Unfortunately, there are a lot of different ways in which energy can be stored, and that can make potential energy very difficult to recognize. In general, an object has potential energy because of its position relative to another object . For example, when you hold a rock above the earth, it has potential energy because of its position relative to the ground. You can tell that this is potential energy because the energy is stored for as long as you hold the rock in the air. Once you drop the rock, though, the stored energy is converted to kinetic energy as the rock falls. A ball at the top of a hill stores potential energy until it is allowed to roll to the bottom. When you hold two magnets next to each other, they store potential energy too. For some examples of potential energy, it is harder to see how "position" is involved.
Chemical Energy
In chemistry, we are often interested in what is called chemical potential energy . Chemical potential energy is energy stored in the atoms, molecules, and chemical bonds that make up matter. How does this depend on position?
As you learned earlier, the world, and all of the chemicals in it are made up of atoms and molecules. These store potential energy that is dependent on their positions relative to one another. Of course, you cannot see atoms and molecules. However, scientists do know a lot about ways in which atoms and molecules interact. This allows them to figure out how much potential energy is stored in a specific quantity (like a cup or a gallon) of a particular chemical. Different chemicals have different amounts of potential energy because they are made up of different atoms, and those atoms have different positions relative to one another.
Since different chemicals have different amounts of potential energy, scientists will sometimes say potential energy depends not only on position , but also on composition . Composition affects potential energy because it determines which molecules and atoms end up next to each other. For example, the total potential energy in a cup of pure water is different than the total potential energy in a cup of apple juice, because the cup of water and the cup of apple juice are composed of different amounts of different chemicals.
At this point, you may be wondering exactly how useful chemical potential energy can be. If you want to release the potential energy stored in an object held above the ground, you just drop it. But how do you get potential energy out of chemicals? It is actually not that difficult. You use the fact that different chemicals have different amounts of potential energy . If you start with chemicals that have a lot of potential energy and allow them to react and form chemicals with less potential energy, all the extra energy that was in the chemicals at the beginning, but not at the end, is released.
Units of Energy
Energy is measured in one of two common units: the calorie and the joule. The joule \(\left( \text{J} \right)\) is the SI unit of energy. The calorie is familiar because it is commonly used when referring to the amount of energy contained within food. A calorie \(\left( \text{cal} \right)\) is the quantity of heat required to raise the temperature of 1 gram of water by \(1^\text{o} \text{C}\). For example, raising the temperature of \(100 \: \text{g}\) of water from \(20^\text{o} \text{C}\) to \(22^\text{o} \text{C}\) would require 100 g x 2 o C = 200 cal.
Calories contained within food are actually kilocalories \(\left( \text{kcal} \right)\). In other words, if a certain snack contains 85 food calories, it actually contains \(85 \: \text{kcal}\) or \(85,000 \: \text{cal}\). In order to make the distinction, the dietary calorie is written with a capital C.
\[1 \: \text{kilocalorie} = 1 \: \text{Calorie} = 1000 \: \text{calories}\]
To say that the snack "contains" 85 Calories means that \(85 \: \text{kcal}\) of energy are released when that snack is processed by your body.
Heat changes in chemical reactions are typically measured in joules rather than calories. The conversion between a joule and a calorie is shown below.
1 cal=4.184 J
We can calculate the amount of heat released in kilojoules when a 400 Calorie hamburger is digested.
\[400 \: \text{Cal} = 400 \: \text{kcal} \times \dfrac{4.184 \: \text{kJ}}{1 \: \text{kcal}} = 1.67 \times 10^3 \: \text{kJ}\]
Summary
- Any time we use energy, we transfer energy from one object to another.
- Energy can be transferred in one of two ways: as heat, or as work.
- Heat is the term given to energy that is transferred from a hot object to a cooler object due to the difference in their temperatures.
- Work is the term given to energy that is transferred as a result of a force applied over a distance.
- Energy comes in two fundamentally different forms: kinetic energy and potential energy.
- Kinetic energy is the energy of motion.
- Potential energy is stored energy that depends on the position of an object relative to another object.
- Chemical potential energy is a special type of potential energy that depends on the positions of different atoms and molecules relative to one another.
- Chemical potential energy can also be thought of as depending on chemical composition.
- Energy can be converted from one form to another. The total amount of mass and energy in the universe is conserved.
Contributions & Attributions
This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:
- Wikibooks
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Henry Agnew (UC Davis)


1: Electric Charges and Fields
In this chapter, we begin the study of the electric force, which acts on all objects with a property called charge. The electric force is much stronger than gravity (in most systems where both appear), but it can be a force of attraction or a force of repulsion, which leads to very different effects on objects. The electric force helps keep atoms together, so it is of fundamental importance in matter. But it also governs most everyday interactions we deal with, from chemical interactions to biological processes.
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- 1.1: Prelude to Electric Charges and Fields
- Back when we were studying Newton’s laws, we identified several physical phenomena as forces. We did so based on the effect they had on a physical object: Specifically, they caused the object to accelerate. Later, when we studied impulse and momentum, we expanded this idea to identify a force as any physical phenomenon that changed the momentum of an object. In either case, the result is the same: We recognize a force by the effect that it has on an object.
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- 1.2: Electric Charge
- You are certainly familiar with electronic devices that you activate with the click of a switch, from computers to cell phones to television. And you have certainly seen electricity in a flash of lightning during a heavy thunderstorm. But you have also most likely experienced electrical effects in other ways, maybe without realizing that an electric force was involved. Let’s take a look at some of these activities and see what we can learn from them about electric charges and forces.
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- 1.3: Conductors, Insulators, and Charging by Induction
- In the preceding section, we said that scientists were able to create electric charge only on nonmetallic materials and never on metals. To understand why this is the case, you have to understand more about the nature and structure of atoms. In this section, we discuss how and why electric charges do—or do not—move through materials. A more complete description is given in a later chapter.
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- 1.4: Coulomb's Law
- Experiments with electric charges have shown that if two objects each have electric charge, then they exert an electric force on each other. The magnitude of the force is linearly proportional to the net charge on each object and inversely proportional to the square of the distance between them. (Interestingly, the force does not depend on the mass of the objects.) The direction of the force vector is along the imaginary line joining the two objects and is dictated by the signs of the charges.
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- 1.6: Calculating Electric Fields of Charge Distributions
- The charge distributions we have seen so far have been discrete: made up of individual point particles. This is in contrast with a continuous charge distribution, which has at least one nonzero dimension. If a charge distribution is continuous rather than discrete, we can generalize the definition of the electric field. We simply divide the charge into infinitesimal pieces and treat each piece as a point charge.
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- 1.7: Electric Field Lines
- Our model is that the charge on an object (the source charge) alters space in the region around it in such a way that when another charged object (the test charge) is placed in that region of space, that test charge experiences an electric force. The concept of electric field lines, and of electric field line diagrams, enables us to visualize the way in which the space is altered, allowing us to visualize the field.
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- 1.8: Electric Dipoles
- Earlier we discussed, and calculated, the electric field of a dipole: two equal and opposite charges that are “close” to each other. (In this context, “close” means that the distance d between the two charges is much, much less than the distance of the field point P, the location where you are calculating the field.) Let’s now consider what happens to a dipole when it is placed in an external field.
Thumbnail: The eight source charges each apply a force on the single test charge Q. Each force can be calculated independently of the other seven forces. This is the essence of the superposition principle.


5: Gases
In this chapter, we examine the relationships between gas temperature, pressure, amount, and volume. We will study a simple theoretical model and use it to analyze the experimental behavior of gases. The results of these analyses will show us the limitations of the theory and how to improve on it.
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- 5.1: Gas Pressure, Temperature, Volume, and Chemical Amount
- Gases exert pressure, which is force per unit area. The pressure of a gas may be expressed in the SI unit of pascal or kilopascal, as well as in many other units including torr, atmosphere, and bar. Atmospheric pressure is measured using a barometer; other gas pressures can be measured using one of several types of manometers.
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- 5.2: Relating Pressure, Volume, Amount, and Temperature - The Ideal Gas Law
- The behavior of gases can be described by several laws based on experimental observations of their properties. including Amontons’s law, Charles’s law, Boyle’s law and Avogadro’s law. These laws can be extracted directly from the ideal gas law.
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- 5.3: Stoichiometry of Gaseous Substances, Mixtures, and Reactions
- The ideal gas law can be used to derive a number of convenient equations relating directly measured quantities to properties of interest for gaseous substances and mixtures. Appropriate rearrangement of the ideal gas equation may be made to permit the calculation of gas densities and molar masses. Dalton’s law of partial pressures may be used to relate measured gas pressures for gaseous mixtures to their compositions.
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- 5.4: The Kinetic-Molecular Theory, Effusion, and Diffusion
- The kinetic molecular theory is a simple but very effective model that effectively explains ideal gas behavior. The theory assumes that gases consist of widely separated molecules of negligible volume that are in constant motion, colliding elastically with one another and the walls of their container with average velocities determined by their absolute temperatures. The individual molecules of a gas exhibit a range of velocities.
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- 5.5: Non-Ideal Gas Behavior
- Gas molecules possess a finite volume and experience forces of attraction for one another. Consequently, gas behavior is not necessarily described well by the ideal gas law. Under conditions of low pressure and high temperature, these factors are negligible, the ideal gas equation is an accurate description of gas behavior, and the gas is said to exhibit ideal behavior. The van der Waals equation is a modified version of the ideal gas law that can be used to account for the non-ideal behavior.
Contributors and Attributions
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).
- Thumbnail: As long as black-body radiation (not shown) doesn't escape a system, atoms in thermal agitation undergo essentially elastic collisions. On average, two atoms rebound from each other with the same kinetic energy as before a collision. Five atoms are colored red so their paths of motion are easier to see. (Public Domain; Greg L via Wikipedia )