3.1: Exponential Growth and Decay
- What does it mean to say that a function is “exponential”?
- How much data do we need to know in order to determine the formula for an exponential function?
- Are there important trends that all exponential functions exhibit?
Linear functions have constant average rate of change and model many important phenomena. In other settings, it is natural for a quantity to change at a rate that is proportional to the amount of the quantity present. For instance, whether you put $\(100\) or $\(100000\) or any other amount in a mutual fund, the investment's value changes at a rate proportional the amount present. We often measure that rate in terms of the annual percentage rate of return.
Suppose that a certain mutual fund has a \(10\)% annual return. If we invest $\(100\text{,}\) after \(1\) year we still have the original $\(100\text{,}\) plus we gain \(10\)% of $\(100\text{,}\) so
If we instead invested $\(100000\text{,}\) after \(1\) year we again have the original $\(100000\text{,}\) but now we gain \(10\)% of $\(100000\text{,}\) and thus
We therefore see that regardless of the amount of money originally invested, say \(P\text{,}\) the amount of money we have after \(1\) year is \(1.1P\text{.}\)
If we repeat our computations for the second year, we observe that
The ideas are identical with the larger dollar value, so
and we see that if we invest \(P\) dollars, in \(2\) years our investment will grow to \(1.1^2 P\text{.}\)
Of course, in \(3\) years at \(10\)%, the original investment \(P\) will have grown to \(1.1^3 P\text{.}\) Here we see a new kind of pattern developing: annual growth of \(10\)% is leading to powers of the base \(1.1\text{,}\) where the power to which we raise \(1.1\) corresponds to the number of years the investment has grown. We often call this phenomenon exponential growth .
Suppose that at age \(20\) you have $\(20000\) and you can choose between one of two ways to use the money: you can invest it in a mutual fund that will, on average, earn \(8\)% interest annually, or you can purchase a new automobile that will, on average, depreciate \(12\)% annually. Let's explore how the $\(20000\) changes over time.
Let \(I(t)\) denote the value of the $\(20000\) after \(t\) years if it is invested in the mutual fund, and let \(V(t)\) denote the value of the automobile \(t\) years after it is purchased.
- Determine \(I(0)\text{,}\) \(I(1)\text{,}\) \(I(2)\text{,}\) and \(I(3)\text{.}\)
- Note that if a quantity depreciates \(12\)% annually, after a given year, \(88\)% of the quantity remains. Compute \(V(0)\text{,}\) \(V(1)\text{,}\) \(V(2)\text{,}\) and \(V(3)\text{.}\)
- Based on the patterns in your computations in (a) and (b), determine formulas for \(I(t)\) and \(V(t)\text{.}\)
- Use Desmos to define \(I(t)\) and \(V(t)\text{.}\) Plot each function on the interval \(0 \le t \le 20\) and record your results on the axes in Figure \(\PageIndex{1}\) , being sure to label the scale on the axes. What trends do you observe in the graphs? How do \(I(20)\) and \(V(20)\) compare?
Exponential functions of form \(f(t) = ab^t\)
In Preview Activity \(\PageIndex{1}\) , we encountered the functions \(I(t)\) and \(V(t)\) that had the same basic structure. Each can be written in the form \(g(t) = ab^t\) where \(a\) and \(b\) are positive constants and \(b \ne 1\text{.}\) Based on our earlier work with transformations, we know that the constant \(a\) is a vertical scaling factor, and thus the main behavior of the function comes from \(b^t\text{,}\) which we call an “exponential function”.
Let \(b\) be a real number such that \(b \gt 0\) and \(b \ne 1\text{.}\) We call the function defined by
an exponential function with base \(b\text{.}\)
For an exponential function \(f(t) = b^t\text{,}\) we note that \(f(0) = b^0 = 1\text{,}\) so an exponential function of this form always passes through \((0,1)\text{.}\) In addition, because a positive number raised to any power is always positive (for instance, \(2^{10} = 1096\) and \(2^{-10} = \frac{1}{2^{10}} = \frac{1}{2096}\)), the output of an exponential function is also always positive. In particular, \(f(t) = b^t\) is never zero and thus has no \(x\)-intercepts.
Because we will be frequently interested in functions such as \(I(t)\) and \(V(t)\) with the form \(ab^t\text{,}\) we will also refer to functions of this form as “exponential”, understanding that technically these are vertical stretches of exponential functions according to Definition 3.1.2. In Preview Activity 3.1.1, we found that \(I(t) = 20000(1.08)^t\) and \(V(t) = 20000(0.88)^t\text{.}\) It is natural to call \(1.08\) the “growth factor” of \(I\) and similarly \(0.88\) the growth factor of \(V\text{.}\) In addition, we note that these values stem from the actual growth rates: \(0.08\) for \(I\) and \(-0.12\) for \(V\text{,}\) the latter being negative because value is depreciating. In general, for a function of form \(f(t) = ab^t\text{,}\) we call \(b\) the growth factor . Moreover, if \(b = 1+r\text{,}\) we call \(r\) the growth rate . Whenever \(b \gt 1\text{,}\) we often say that the function \(f\) is exhibiting “exponential growth”, wherease if \(0 \lt b \lt 1\text{,}\) we say \(f\) exhibits “exponential decay”.
We explore the properties of functions of form \(f(t) = ab^t\) further in Activity \(\PageIndex{2}\)
In Desmos , define the function \(g(t) = ab^t\) and create sliders for both \(a\) and \(b\) when prompted. Click on the sliders to set the minimum value for each to \(0.1\) and the maximum value to \(10\text{.}\) Note that for \(g\) to be an exponential function, we require \(b \ne 1\text{,}\) even though the slider for \(b\) will allow this value.
- What is the domain of \(g(t) = ab^t\text{?}\)
- What is the range of \(g(t) = ab^t\text{?}\)
- What is the \(y\)-intercept of \(g(t) = ab^t\text{?}\)
- How does changing the value of \(b\) affect the shape and behavior of the graph of \(g(t) = ab^t\text{?}\) Write several sentences to explain.
- For what values of the growth factor \(b\) is the corresponding growth rate positive? For which \(b\)-values is the growth rate negative?
- Consider the graphs of the exponential functions \(p\) and \(q\) provided in Figure 3.1.3. If \(p(t) = ab^t\) and \(q(t) = cd^t\text{,}\) what can you say about the values \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) (beyond the fact that all are positive and \(b \ne 1\) and \(d \ne 1\))? For instance, can you say a certain value is larger than another? Or that one of the values is less than \(1\text{?}\)
Determining formulas for exponential functions
To better understand the roles that \(a\) and \(b\) play in an exponential function, let's compare exponential and linear functions. In Table \(\PageIndex{4}\) and Table \(\PageIndex{5}\) , we see output for two different functions \(r\) and \(s\) that correspond to equally spaced input.
| \(t\) | \(0\) | \(3\) | \(6\) | \(9\) |
| \(r(t)\) | \(12\) | \(10\) | \(8\) | \(6\) |
| \(t\) | \(0\) | \(3\) | \(6\) | \(9\) |
| \(s(t)\) | \(12\) | \(9\) | \(6.75\) | \(5.0625\) |
In Table \(\PageIndex{4}\) , we see a function that exhibits constant average rate of change since the change in output is always \(\triangle r = -2\) for any change in input of \(\triangle t = 3\text{.}\) Said differently, \(r\) is a linear function with slope \(m = -\frac{2}{3}\text{.}\) Since its \(y\)-intercept is \((0,12)\text{,}\) the function's formula is \(y = r(t) = 12 - \frac{2}{3}t\text{.}\)
In contrast, the function \(s\) given by Table \(\PageIndex{5}\) does not exhibit constant average rate of change. Instead, another pattern is present. Observe that if we consider the ratios of consecutive outputs in the table, we see that
So, where the differences in the outputs in Table \(\PageIndex{4}\) are constant, the ratios in the outputs in Table 3.1.5 are constant. The latter is a hallmark of exponential functions and may be used to help us determine the formula of a function for which we have certain information.
If we know that a certain function is linear, it suffices to know two points that lie on the line to determine the function's formula. It turns out that exponential functions are similar: knowing two points on the graph of a function known to be exponential is enough information to determine the function's formula. In the following example, we show how knowing two values of an exponential function enables us to find both \(a\) and \(b\) exactly.
Suppose that \(p\) is an exponential function and we know that \(p(2) = 11\) and \(p(5) = 18\text{.}\) Determine the exact values of \(a\) and \(b\) for which \(p(t) = ab^t\text{.}\)
- Solution
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Since we know that \(p(t) = ab^t\text{,}\) the two data points give us two equations in the unknowns \(a\) and \(b\text{.}\) First, using \(t = 2\text{,}\)
\[ ab^2 = 11\text{,}\label{eq-growth-p-2}\tag{3.1.1} \]and using \(t = 5\) we also have
\[ ab^5 = 18\text{.}\label{eq-growth-p-5}\tag{3.1.2} \]Because we know that the quotient of outputs of an exponential function corresponding to equally-spaced inputs must be constant, we thus naturally consider the quotient \(\frac{18}{11}\text{.}\) Using Equation (3.1.1) and Equation (3.1.2), it follows that
\[ \frac{18}{11} = \frac{ab^5}{ab^2}\text{.} \nonumber \]Simplifying the fraction on the right, we see that \(\frac{18}{11} = b^3 \text{.}\) Solving for \(b\text{,}\) we find that \(b = \sqrt[3]{\frac{18}{11}}\) is the exact value of \(b\text{.}\) Substituting this value for \(b\) in Equation (3.1.1), it then follows that \(a \left( \sqrt[3]{\frac{18}{11}} \right)^2 = 11\text{,}\) so \(a = \frac{11}{\left( \frac{18}{11} \right)^{2/3}} \text{.}\) Therefore,
\[ p(t) = \frac{11}{\left( \frac{18}{11} \right)^{2/3}} \left( \sqrt[3]{\frac{18}{11}} \right)^t \approx 7.9215 \cdot 1.1784^t\text{,} \nonumber \]and a plot of \(y = p(t)\) confirms that the function indeed passes through \((2,11)\) and \((5,18)\) as shown in Figure \(\PageIndex{7}\)
The value of an automobile is depreciating. When the car is \(3\) years old, its value is $\(12500\text{;}\) when the car is \(7\) years old, its value is $\(6500\text{.}\)
- Suppose the car's value \(t\) years after its purchase is given by the function \(V(t)\) and that \(V\) is exponential with form \(V(t) = ab^t\text{,}\) what are the exact values of \(a\) and \(b\text{?}\)
- Using the exponential model determined in (a), determine the purchase value of the car and estimate when the car will be worth less than $\(1000\text{.}\)
- Suppose instead that the car's value is modeled by a linear function \(L\) and satisfies the values stated at the outset of this activity. Find a formula for \(L(t)\) and determine both the purchase value of the car and when the car will be worth $\(1000\text{.}\)
- Which model do you think is more realistic? Why?
Trends in the behavior of exponential functions
Recall that a function is increasing on an interval if its value always increases as we move from left to right. Similarly, a function is decreasing on an interval provided that its value always decreases as we move from left to right.
An additional trend is apparent in the graphs in Figure \(\PageIndex{8}\) and Figure \(\PageIndex{9}\) . Each graph bends upward and is therefore concave up. We can better understand why this is so by considering the average rate of change of both \(f\) and \(g\) on consecutive intervals of the same width. We choose adjacent intervals of length \(1\) and note particularly that as we compute the average rate of change of each function on such intervals,
Thus, these average rates of change are also measuring the total change in the function across an interval that is \(1\)-unit wide. We now assume that \(f(t) = 2 (1.25)^t\) and \(g(t) = 8(0.75)^t\) and compute the rate of change of each function on several consecutive intervals.
| \(t\) | \(f(t)\) | \(AV_{[t,t+1]}\) |
| \(0\) | \(2\) | \(0.5\) |
| \(1\) | \(2.5\) | \(0.625\) |
| \(2\) | \(3.125\) | \(0.78215\) |
| \(3\) | \(3.90625\) | \(0.97656\) |
| \(t\) | \(g(t)\) | \(AV_{[t,t+1]}\) |
| \(0\) | \(8\) | \(-2\) |
| \(1\) | \(6\) | \(-1.5\) |
| \(2\) | \(4.5\) | \(-1.125\) |
| \(3\) | \(3.375\) | \(-0.84375\) |
From the data in Table \(\PageIndex{10}\) , we see that the average rate of change is increasing as we increase the value of \(t\text{.}\) We naturally say that \(f\) appears to be “increasing at an increasing rate”. For the function \(g\text{,}\) we first notice that its average rate of change is always negative, but also that the average rate of change gets less negative as we increase the value of \(t\text{.}\) Said differently, the average rate of change of \(g\) is also increasing as we increase the value of \(t\text{.}\) Since \(g\) is always decreasing but its average rate of change is increasing, we say that \(g\) appears to be “decreasing at an increasing rate”. These trends hold for exponential functions generally 1 according to the following conditions.
For an exponential function of the form \(f(t) = ab^t\) where \(a\) and \(b\) are both positive with \(b \ne 1\text{,}\)
- if \(b \gt 1\text{,}\) then \(f\) is always increasing and always increases at an increasing rate;
- if \(0 \lt b \lt 1\text{,}\) then \(f\) is always decreasing and always decreases at an increasing rate.
Observe how a function's average rate of change helps us classify the function's behavior on an interval: whether the average rate of change is always positive or always negative on the interval enables us to say if the function is always increasing or always decreasing, and then how the average rate of change itself changes enables us to potentially say how the function is increasing or decreasing through phrases such as “decreasing at an increasing rate”.
For each of the following prompts, give an example of a function that satisfies the stated characteristics by both providing a formula and sketching a graph.
- A function \(p\) that is always decreasing and decreases at a constant rate.
- A function \(q\) that is always increasing and increases at an increasing rate.
- A function \(r\) that is always increasing for \(t \lt 2\text{,}\) always decreasing for \(t \gt 2\text{,}\) and is always changing at a decreasing rate.
- A function \(s\) that is always increasing and increases at a decreasing rate. (Hint: to find a formula, think about how you might use a transformation of a familiar function.)
- A function \(u\) that is always decreasing and decreases at a decreasing rate.
Summary
- We say that a function is exponential whenever its algebraic form is \(f(t) = ab^t\) for some positive constants \(a\) and \(b\) where \(b \ne 1\text{.}\) (Technically, the formal definition of an exponential function is one of form \(f(t) = b^t\text{,}\) but in our everyday usage of the term “exponential” we include vertical stretches of these functions and thus allow \(a\) to be any positive constant, not just \(a = 1\text{.}\))
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To determine the formula for an exponential function of form \(f(t) = ab^t\text{,}\) we need to know two pieces of information. Typically this information is presented in one of two ways.
- If we know the amount, \(a\text{,}\) of a quantity at time \(t = 0\) and the rate, \(r\text{,}\) at which the quantity grows or decays per unit time, then it follows \(f(t) = a(1+r)^t\text{.}\) In this setting, \(r\) is often given as a percentage that we convert to a decimal (e.g., if the quantity grows at a rate of \(7\)% per year, we set \(r = 0.07\text{,}\) so \(b = 1.07\)).
- If we know any two points on the exponential function's graph, then we can set up a system of two equations in two unknowns and solve for both \(a\) and \(b\) exactly. In this situation, it is useful to consider the quotient of the two known outputs, as demonstrated in Example 3.1.6.
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Exponential functions of the form \(f(t) = ab^t\) (where \(a\) and \(b\) are both positive and \(b \ne 1\)) exhibit the following important characteristics:
- The domain of any exponential function is the set of all real numbers and the range of any exponential function is the set of all positive real numbers.
- The \(y\)-intercept of the exponential function \(f(t) = ab^t\) is \((0,a)\) and the function has no \(x\)-intercepts.
- If \(b \gt 1\text{,}\) then the exponential function is always increasing and always increases at an increasing rate. If \(0 \lt b \lt 1\text{,}\) then the exponential function is always decreasing and always decreases at an increasing rate.
Exercises
Grinnell Glacier in Glacier National Park in Montana covered about \(142\) acres in 2007 and was found to be shrinking at about \(4.4\)% per year. 2
- Let \(G(t)\) denote the area of Grinnell Glacier in acres in year \(t\text{,}\) where \(t\) is the number of years since 2007. Find a formula for \(G(t)\) and define the function in Desmos .
- How many acres of ice were in the glacier in 1997? In 2012? What does the model predict for 2022?
- How many total acres of ice were lost from 2007 to 2012?
- What was the average rate of change of \(G\) from 2007 to 2012? Write a sentence to explain the meaning of this number and include units on your answer. In addition, how does this compare to the average rate of change of \(G\) from 2012 to 2017?
- How would you you describe the overall behavior of \(G\text{,}\) and thus what is happening to the Grinnell Glacier?
Consider the exponential function \(f\) whose graph is given by Figure \(\PageIndex{12}\) Note that \(f\) passes through the two noted points exactly.
- Determine the values of \(a\) and \(b\) exactly.
- Determine the average rate of change of \(f\) on the intervals \([2,7]\) and \([7,12]\text{.}\) Which average rate is greater?
- Find the equation of the linear function \(L\) that passes through the points \((2,20)\) and \((7,5)\text{.}\)
- Which average rate of change is greater? The average rate of change of \(f\) on \([0,2]\) or the average rate of change of \(L\) on \([0,2]\text{?}\)
A cup of hot coffee is brought outside on a cold winter morning in Winnipeg, Manitoba, where the surrounding temperature is \(0\) degrees Fahrenheit. A temperature probe records the coffee's temperature (in degrees Fahrenheit) every minute and generates the data shown in Table \(\PageIndex{13}\)
| \(t\) | \(0\) | \(2\) | \(4\) | \(6\) | \(8\) | \(10\) |
| \(F(t)\) | \(175\) | \(129.64\) | \(96.04\) | \(71.15\) | \(52.71\) | \(39.05\) |
- Assume that the data in the table represents the overall trend of the behavior of \(F\text{.}\) Is \(F\) linear, exponential, or neither? Why?
- Is it possible to determine an exact formula for \(F\text{?}\) If yes, do so and justify your formula; if not, explain why not.
- What is the average rate of change of \(F\) on \([4,6]\text{?}\) Write a sentence that explains the practical meaning of this value in the context of the overall exercise.
- How do you think the data would appear if instead of being in a regular coffee cup, the coffee was contained in an insulated mug?
The amount (in milligrams) of a drug in a person's body following one dose is given by an exponential decay function. Let \(A(t)\) denote the amount of drug in the body at time \(t\) in hours after the dose was taken. In addition, suppose you know that \(A(3) = 22.7\) and \(A(6) = 15.2\text{.}\)
- Find a formula for \(A\) in the form \(A(t) = ab^t\text{,}\) where you determine the values of \(a\) and \(b\) exactly.
- What is the size of the initial dose the person was given?
- How much of the drug remains in the person's body \(8\) hours after the dose was taken?
- Estimate how long it will take until there is less than \(1\) mg of the drug remaining in the body.
- Compute the average rate of change of \(A\) on the intervals \([3,5]\text{,}\) \([5,7]\text{,}\) and \([7,9]\text{.}\) Write at least one careful sentence to explain the meaning of the values you found, including appropriate units. Then write at least one additional sentence to explain any overall trend(s) you observe in the average rate of change.
- Plot \(A(t)\) on an appropriate interval and label important points and features of the graph to highlight graphical interpretations of your answers in (b), (c), (d), and (e).
<1> It takes calculus to justify this claim fully and rigorously.


73: Catwoman’s Hyde: A Comparative Reading of the 2002 Catwoman Relaunch and Stevenson’s Novella (Syn)
By Lesa Syn
#heroes #review #academic #analysis #artsandculture #descriptive #scholarly
Abstract
This article advocates that the comic character of Catwoman is a comic incarnation of Robert Louis Stevenson’s Edward Hyde. It does this first by problematizing Andreas Reichstein’s reading of Batman as Hyde (1998). While the similarity between Bruce Wayne and Dr. Henry Jekyll is considerable (such as both being accomplished and affluent men of science who have nocturnal alter egos), Hyde embodies hedonistic desire and loss of control while Batman is the incarnation of discipline and control. This work then goes on to offer the numerous and stark similarities between Hyde and Catwoman, such as offering their counterparts animalistic freedom and the ability to achieve unification through embracing their darker halves. Because of her desire to embrace her dual human experience, the hero/villain Catwoman encapsulates the most human of comic characters.
Keywords: Catwoman , Batman , Hulk , Henry Jekyll , Edward Hyde , Ed Brubaker , Darwin Cooke , Robert Louis Stevenson , Andreas Reichstein , Identity Split , Hero/Villain , animalistic
Who is stronger, Catwoman or Hulk? In stark contrast to the Hulk, who is arguably the strongest comic-book character ever, Catwoman—like her beloved Batman—is one of the few comic-book characters that is “not super-powered, an alien, or a mutant,” but rather entirely human (Orr 1984: 176–177). Well, almost entirely human. Therefore, comparing Catwoman to Hulk may as well be comparing a person to a natural disaster; however, Jason Ranker (2008) discusses a comparison of these dissimilar comic characters… and, believe it or not, the comparison is apt.
In his research, Ranker highlights how Catwoman and Hulk represent stereotypical versions of strength constructed along gender lines; however, there is a better reason to place these two in a category of their own. Unlike almost every other character in the comic book multiverse, these two characters are both heroes and villains, often at the same time. Hulk continues to be one of the greatest threats to Earth but he is also a founding member of The Avengers. Similarly, Catwoman is a villain on par with the rest of Forever Evil, but she is also a founding member of the new Justice League of America (see Figure 1 ): “Unfortunately, being a fence-sitter on the thin line separating good and evil doesn’t make her a neutral party” ( Beatty 2004: 36 ). These are ambivalent characters that are neither absolute good nor evil, but are constantly torn between both aspects—they are both modern and powerful Edward Hydes.
Perhaps it can be argued that several villains have aspects similar to Robert Louis Stevenson’s iconic Edward Hyde. Two-Face is a literal Janus figure and often physically depicts both good and evil, and Clayface’s inability to fully control his transformation often results in Hyde-like horror in those around him. However, Two-Face does not suffer from his ambivalence, but simply relegates his dual nature to the fate of a coin flip. Clayface is not noble, like Jekyll, but rather is a disguised monster. Catwoman and her alter ego Selina Kyle, torn between both evil and good, are the only true Hyde and Jekyll of Gotham.
Adam Capitainio ( 2010 ) has thoroughly compared Marvel’s Bruce Banner and the Incredible Hulk to Stevenson’s Dr. Jekyll and Mr. Hyde, so this paper will move the discussion to the similar intellectual and emotional torment of DC’s Kyle and Catwoman. To accomplish this task, this paper will first problematize Andreas Reichstein’s (1988) reading of Batman as Mr. Hyde and then it will offer tangible evidence—especially from Brubaker, Cooke, and Allard’s 2002 Catwoman relaunch—that, despite the physical dissimilarities of Kyle to Jekyll, Catwoman is a more complete version of a modern day Mr. Hyde.
Is Batman an American Mr. Hyde?
Reichstein ( 1998 ) makes a compelling argument that Batman’s alter ego Bruce Wayne is an American Jekyll. Beyond the obvious comparisons of the successful, popular, and affluent bachelors, these are men of prominence and success who only transform into their alter egos at night. Reichstein highlights that despite both men being childless, Wayne and Jekyll have paternal affection for others: Wayne cares for his various wards and “Jekyll confesses how he had a father’s concern for Hyde” ( Reichstein 1998: 340 ). Reichstein also discusses how both Wayne and Jekyll spend the vast majority of their free time in their secret laboratories, conducting various chemical experiments.
However, the staunchest similarity between these two characters is their double lives. Reichstein explains, “Besides all these formal similarities between Wayne and Jekyll, the essential link between these characters lies in their basic trait: their double identity, their double personality” ( Reichstein 1998: 343 ). As Philip Orr discusses, “[When] Bruce Wayne refers to the Batman, he is not referring to himself in the third person; rather he is referring to the other” (Orr 1995: 174). Rather than just one person with two aspects, Wayne and Batman are two distinct individuals who just happen to exist within a common body. Wayne even confesses this idea to a psychologist: “I guess, we’re all two people—one in daylight and one we keep in shadow” (Batman Returns). Similarly, in his final confession, Dr. Jekyll writes, “With every day, and from both sides of my intelligence, the moral and the intellectual, I thus drew steadily nearer to that truth … that man is not truly one, but truly two” ( Stevenson 1886: 77 ). Clearly, both of these men of science are all too intimately familiar with the dual nature of humanity, and their polarizing personalities. However, it is worth highlighting that the parallels between these two characters lie almost exclusively along the similarities of Wayne and Jekyll.
As such, while Wayne is an American Jekyll, what is the answer to Reichstein’s ( 1998 ) titular question “Batman—an American Mr. Hyde?” Both Batman and Hyde are empowered shadows of their other selves, and are able to accomplish what neither Wayne nor Jekyll ever could. They achieve this power by embracing animalistic aspects of their personalities. Jekyll describes his other self as playing “ape-like tricks” ( Stevenson 1886: 91 ) and possessing “ape-like spite” (93). Capitainio explains, “This suggests that Hyde, as the hidden side of Jekyll’s personality, is representative of the animal past and behavior that human beings have necessarily repressed in their quest for ‘civilization’” ( Capitainio 2010: 250 ).
While Jekyll is an accomplished pinnacle of human evolution—generous, brilliant, scientific, attractive, and affluent—Hyde returns this character to his evolutionary ape-like past, which was a common theme in the Gothic novels of the 1890s ( Reichstein 1998: 346 ). Regarding the newfound freedom and anonymity of becoming Hyde, Jekyll writes, “And thus fortified, as I supposed, on every side, I began to profit by the strange immunities of my position” ( Stevenson 1886: 86 ). Batman too is able to tap into his primal animal form—in this case a bat—and uses this animal aspect to achieve his deepest desire for vengeance against the criminal element that orphaned him as a child. Batman also allowed Wayne to safely fulfill his desires for vigilantism.
While such similarities exist between Hyde and Batman, one of the most interesting parallels between these two characters is that both of them murder adversaries in their initial appearances. Mr. Hyde brutally assaults Sir Danvers Carew after a perceived insult: “And next moment, with ape-like fury, he was trampling his victim under foot and hailing down a storm of blows, under which the bones were audibly shattered and the body jumped upon the roadway” ( Stevenson 1886: 27 ). Similarly, in Batman’s first appearance, he is holding the villainous Stryker when “… suddenly, Stryker, with the strength of a madman, tears himself free from the grasp of the Bat-Man…” ( Finger and Kane 1939: 6 ). Stryker then pulls out a gun and shoots at Batman, but Batman punches Stryker, knocking him over the rail and into an acid tank, to which Batman comments, “A fitting end for his kind” ( Finger and Kane 1939: 6 ). While both men have murdered an adversary, Hyde has killed an upstanding gentleman and must pay for his action with his life, while Batman has killed a murderer and deserves an accolade. Hyde must hide behind Jekyll or face the gallows, whereas Batman can continue crime-fighting, and be a vigilante without repercussion. Despite the consequences of these actions, Hyde only longs to kill, while Batman resolves to never take a life.
While surface similarities—such as the stark similarities between their alter egos, empowerment through animalistic disguise, and both having taken a life—exist between Hyde and Batman these characters are as dissimilar from each other as they are from their better halves. For Jekyll, becoming Hyde was embracing and savoring his dark side, “I knew myself, at the first breath of this new life, to be more wicked, tenfold more wicked, sold a slave to my original evil; and the thought, in that moment, braced and delighted me like wine” ( Stevenson 1886: 79 ). Jekyll seems to use Hyde to indulge his baser desires. This is completely different from Batman whose “armor/costume harnesses the evil inside and outside of him, as well as protects against repressed sexual desires” ( Reichstein 1998: 348 ). While Hyde is allowed to run rampant, participating in any debauchery that he comes upon, Batman focuses only on punishing criminals.
Most of all, Hyde embodies Jekyll’s willing loss of control, while “Wayne/Batman is control” ( Reichstein 1998: 347 ). After Hyde has killed Carew, Jekyll abandons the potion that unleashed his sinister self, but Hyde continued to reassert himself. Hyde manifests himself at first in dreams and then in reality by forcing the transformation unaided by any chemical concoction. Jekyll could not stop Hyde from taking over their body more and more. Batman, on the other hand, could be taken off at any time, cast aside. Reichstein explains, “He can don the costume/armor whenever he wants and drop it again to become Bruce Wayne” ( Reichstein 1998: 348 ). Reichstein ultimately concludes, “Thus, Batman really is an American cousin of Edward Hyde” ( Reichstein 1998: 350 ). However, this paper suggests that by Reichstein’s definition, Hyde’s familial relationship might as well just as easily include comic book characters such Green Arrow or Iron Man, both of whom are—beneath their masks—affluent playboy bachelor inventors who don costumes/armors to combat evil. While the similarities between Jekyll and Wayne are notable, Batman is not a complete Hyde.
No, but Catwoman Is
However, Catwoman, on the other hand, shares many similarities with Hyde. Jekyll describes the initial transformation into like this: “I felt younger, lighter, happier in body; within I was conscious of a heady recklessness, a current of disordered sensual images running like a millrace in my fancy, a solution of the bonds of obligation, an unknown but not an innocent freedom of the soul” ( Stevenson 1886: 78 ). Kyle too feels the strength and freedom of her alter ego: “That had been one of the reasons for the mask, initially. To help provide. That and the excitement… the adventure. Don’t kid yourself that they weren’t a big part of it, too” ( Brubaker, Cooke, and Allred 2002: 48 ). There is a youthful vibrancy and hedonistic abandonment to both Hyde and Catwoman. Catwoman is unbridled. Throughout Catwoman’s history, she is known as thief, and she ultimately ignores societal rules while helping herself to whatever she desires. Ed Brubaker explains, “She was like Robin Hood, except she forgot to the money to the poor” ( Beatty 2004: 6 ). Catwoman, like Hyde, is in it for the freedom and youthful thrill.
Jekyll and Hyde, Kyle and Catwoman are completely different people from their counterpart selves. Hyde has his own apartment and his own companions that exist in totally different circles than Jekyll. Similarly, as Beatty explains, “Aside from a few friends and lovers, Selina and Catwoman are two different women moving in different worlds. And that suits them both just fine” ( Beatty 2004: 19 ). Jekyll/Hyde and Catwoman/Kyle are in actuality two individuals who share one common body and one common memory, but share little else.
Hyde and Catwoman are impervious shields for their more noble halves. No matter the potential atrocities that Hyde could ever commit, Jekyll believes that at any moment he could dispense Hyde instantly and permanently:
Think of it—I did not even exist! Let me escape into my laboratory door, give me but a second or two to mix and swallow the draught that I had always standing ready; and whatever he had done, Edward Hyde would pass away like the stain of breath upon a mirror; and there in his stead, quietly at home, trimming the midnight lamp in his study, a man who could afford to laugh at suspicion, would be Henry Jekyll. ( Stevenson 1886: 82 ).
Through his potion, Jekyll could transform into his shadow self and fulfill his most monstrous desires. In the same vein, in The Dark End of The Street, Kyle embraces her bestial aspect, throwing off all moral, legal, and earthly limitations by transforming herself into a Cat-Woman. In this comic, The Cat-side is able to beat up her (masculine) enemies, climb tall buildings, and seemingly fly through the air (via her whip).
It is important to note in Catwoman history, she was known as “The Cat” and first appeared in the first issue of Batman in 1940. However, in 1950, ten years after “The Cat’s” inception, the mild-mannered alter ego Kyle was named. It is a common misconception that Kyle came first—not a decade later—but this idea of the villainous side appearing before the sociality acceptable persona parallel’s Stevenson’s novella in which the story of Hyde accosting the young girl appears pages before the discussion of Jekyll. This argument will now go back to a previous quote from Jekyll, “I knew myself, at the first breath of this new life, to be more wicked, tenfold more wicked, sold a slave to my original evil; and the thought, in that moment, braced and delighted me like wine” ( Stevenson 1886: 79 ). Jekyll seems to use Hyde to indulge his baser desires, but in reality the reverse is true—Hyde uses Jekyll for his own desires.
The problem is that these shielding personas eventually took on a life all their own. Hyde lashed out at his alter ego, punishing him through the destruction of Jekyll’s belongings: “Hence the ape-like tricks that he would play me, scrawling in my own hand blasphemies on the pages of my books, burning the letters and destroying the portrait of my father” ( Stevenson 1886: 91 ). In Brubaker, Cooke, and Allread’s comic, Catwoman too began to have a life outside of Kyle, who says, “The mask is part of who I am now. But it’s also part of the problem, too… because it became a person all on its own” ( Brubaker, Cooke, and Allred 2002: 54 ). Thus, while Hyde and Catwoman were created to liberate their alter egos and allowed them to achieve humanly-impossible, animalistic acts, these personas became whole persons in and of themselves. The animal-side uncontrollable and often exhibits behaviors that are outside of normative human roles. Catwoman is wild, something that Kyle struggles to understand (Figure 1 ). Furthermore, in Figure 1 , this Cat-self bides for ownership of the body she cohabitates and eventually becomes intricately integrated and a fully realized being.
Ultimately, Hyde asserts himself and begins replacing Jekyll. The problem is that, with Carew’s murder on his head, Hyde’s only salvation lies in being Jekyll, but he could not condemn himself to perpetual torpor; as such, “Unable to compound the remedy that turns him into Dr. Jekyll again one day, Jekyll/Hyde kills himself” ( Reichstein 1998: 339 ). Similarly, in this comic, Catwoman killed off both Kyle and then herself ( Brubaker, Cooke, and Allred 2002: 16 )… or at least Catwoman went through a lot of trouble to make it appear that way.  ( Brubaker, Cooke, and Allred 2002: 33 ). Kyle reemerges and tries to live again, believing that Catwoman is dead and buried; however, Catwoman too refuses to stay dead. As Orr says, “[Catwoman] won’t be killed” ( Orr 1994: 181 ). Catwoman’s immortality is in fact the sole attribute that Ranker’s (2008) work offers as to why she is stronger than the Hulk. Beatty too notes, “Selina definitely has nine lives considering the number of times she has survived near-fatal catastrophes” ( Beatty 2004: 25 ). While most often the nine lives of the Catwoman are seen in response to her battle with others, this immortality applies to her battle with Kyle as well.
The immortal Catwoman torments Kyle, much as Hyde tortures Jekyll: “And this again, that the insurgent horror was knit to him closer than a wife, closer than an eye; lay caged in his flesh, where he heard it mutter and felt it struggle to be born” (Stevenson 1863: 91). Kyle struggles to control the beast within her, but unlike Jekyll who does not seek to understand Hyde—Kyle grapples with accepting Catwoman as a part of herself. Kyle waits a moment and then answers Bradley’s question: “I don’t know. Hopefully someone who can look in the mirror without any pain” ( Brubaker, Cooke, and Allred 2002: 16 ). In the mirror here, Kyle sees all the pain that she has caused and that she has endured. Jekyll has a similar mirror experience, but his is all the more tragic because of the loss of control that it represents: “[and] bounding from my bed I rushed to the mirror. At the sight that met my eyes, my blood was changed into something exquisitely thin and icy. Yes, I had gone to bed Henry Jekyll, I had awakened Edward Hyde” ( Stevenson 1886: 83 ).
It is important to highlight that mirrors denote the true representation of self. Christian Metz explains how it is only through a mirror that people can understand themselves and construct their egos: “Thus the child’s ego is formed by identification with its like, and this in two senses simultaneously, metonymically and metaphorically: the other human being who is in the glass, the own reflection which is not the body, which is like it” ( Metz 1975: 48–49 ). Metz underscores the duality of co-existence of each individual’s two selves—the rational ego and the primal id. Thus, by a proverbial mirror, Kyle is able to be a full self, and it is in this mirror, that she understands the Cat’s self. Another way to put it is that the animal-self allows Kyle the only way for her survive and to be a full and actualized person. Whereas the rest of humanity strives to deny the animalistic id inside them, Kyle realizes that she ultimately needs to embrace her Cat self into her life.
Whereas Jekyll embraced the concept of humanity’s dual nature, he theorized that each person was in reality a conglomeration of individuals forced confined in uncomfortable cohabitation: “I hazard the guess that man will be ultimately known for a mere polity of multifarious, incongruous and independent denizens” ( Stevenson 1886: 78 ). In her nightmares, Kyle wrestles, not with Catwoman, but with the vast multitude of individuals that exist inside her: “Like everything I’ve ever been is struggling inside me… trying to find some place to fit themselves” ( Brubaker, Cooke, and Allred 2002: 44 ). On the surface, it is easy to focus simply on the duality, but for both Jekyll and Kyle, the double personalities that they battle are just the beginning, as if both these two—and everyone of us for that matter—could just as easily say, “My name is Legion; for we are many” (Mark 5: 9).
While Kyle and Catwoman embody just two persons that inhabit their body, there are multitudes of others. Ignoring the staunch differences in attitude and mannerisms between Kyle and Catwoman, both characters pronouncedly vacillate in skin tone and costume within themselves—even just within this single comic book. For instance, while the panels within the comic showcase a Catwoman who is very overtly a pale Caucasian brunette, the various covers from this same comic depict a darker complected Kyle that is clearly mulatto, Hispanic, or even Italian (see Figure 3 , left side). However, even this ethnic alteration pales to the disguises that Kyle dons that make her unrecognizable, the staunchest of which is when Kyle goes undercover as a blond bombshell (see Figure 3 , right side). Thus comic’s art work is indicative of Kyle and Catwoman’s personality and multiplicity. Despite all of her various incarnations, Kyle and Catwoman are always at their cores women: “She was a ‘woman,’ and all woman at that” ( Madrid 2009: 248 ). Kyle summons up her female alter egos, uses them as tools, and discards them just as quickly. Kyle can control all these incarnations of herself—other than Catwoman.
Kyle’s therapist asks her, “How long has it been now since you put on the [Catwoman] outfit?” to which Kyle replies, “The outfit? Oh, yeah… that. Almost six months” ( Brubaker, Cooke, and Allred 2002: 45 ). Like Jekyll, Kyle has tried to repress her own Hyde by abandoning the hide of Catwoman. Also like Jekyll, Hyde (i.e.: Catwoman) fought back: “[But] I was still cursed with my duality of purpose; and as the first edge of my penitence wore off, the lower side of me, so long indulged, so recently chained down, began to growl for license” ( Stevenson 1886: 87 ). Catwoman plagues Kyle with
hellish dreams in which she is surrounded by Catwomen who set Kyle on fire, burning away her humanity and leaving only the silhouette of the Catwoman (see Figure 2 ). In this suffering, Kyle is like Jekyll: “I began to be tortured with throes and longings, as of Hyde struggling after freedom” ( Stevenson 1886: 85 ). Kyle feels the Cat personality struggling for freedom. However, Kyle tries to avoid the temptation of becoming Catwoman—which is why she is seeing a therapist—but Catwoman refuses to let her get a moment’s peace until Kyle acquiesces. Catwoman refuses to let Kyle be complete until Kyle fully embraces all of her aspects.
Conclusion
Reichstein tries to reduce Catwoman to little more than one of the “mostly bizarre array of villains in the Batman comics, like the Joker, the Penguin, the Catwoman … [who] reflect the purely bad side of Batman—like counterparts, or mirrors showing him what would happen to him if he lost control” ( Reichstein 1998: 346–347 ). However, Catwoman should not be reduced to simply reflect the bad side of Batman. While Bruce Wayne is similar to a Dr. Henry Jekyll, Batman is not a complete Mr. Edward Hyde… but Catwoman is.
This argument suggests that Catwoman is such an integral part of Kyle and that her life is shattered—without her “other.” While other critics have studied Batman and Hulk—no one has considered Catwoman as Hyde. It is interesting to note that Kyle logically and consciously puts on Catwoman and both co-habitat the same space. Again, this is very akin to Jekyll and Hyde, who—although different in body and manner—share the same memory ( Stevenson 1886: 85 ) and handwriting (89). These are two parts of a cohesive whole, both vying for control over their life while staunchly rebelling against being controlled by any man (even a Batman) and any law.
Reichstein succeeds at linking Jekyll with Wayne, but she ignores the complexity of comparing Hyde to Batman. Thus, this work corrected that oversight by advocating Jekyll and Hyde story is about identity structure: the socially acceptable personality versus the uncontrollable and often intolerable. A close examination of Dark End of the Street reveals the identity problems in Stevenson’s novella also manifest in Kyle and Catwoman. This comic reinforced the Jekyll side is about social conformity, whereas the Hyde represents the uncontainable and unacceptable. Thus, this paper argues that offering a critical approach to Catwoman as Hyde is the only way for freedom from social norms and offers independence from social conformity.
It is noteworthy that Kyle’s Hyde self is not about a tortured individual who seeks to redeem herself from her alter ego, but about accepting that separate self and releasing it from the confines of societal rules. In other words, Catwoman is the ability for humankind to negotiate life with a wild animalistic side. Catwoman’s fictional personality offers readers a chance to enter a new sphere of identity understanding. However, instead of untamed mayhem or total anarchy, Catwoman tests the limits of what life would be like without the rules and limitations of social rules. The tragedy of Stevenson’s novella is not the fall of a celebrity, but rather the fall of one of us. Jekyll is a friend and peer, not an estranged and obsessive recluse like Wayne. As such, Kyle is more like Jekyll—like all of us—in that she is a complex character torn between manifestations of good and evil. Catwoman, the Hyde-like heroic villain and villainous hero, and is one of the most humanistic comic characters of all.
Acknowledgements
The author would like to thank Benjamin Syn for his continuous support and dedication to this project. This project would have been an impossible task without the help that Benjamin gave during the extent of the project. As this project has taken almost a year from topic idea to formulating a well-crafted research paper—Benjamin has been a vital aspect to the project. He often brainstormed, gave relevant insights into my argument, and often offered substantive feedback to this paper. Thank you Benjamin Syn for being an ongoing partner in my quest to finish my degree and my obnoxious desire to become a published writer. It is only because of Benjamin’s dedication—that I am who I am…I adore you and I thank you.
References
- Beatty, S (2004). Catwoman: The Visual Guide to the Feline Fatale. London: Dorling Kindersley.
- Brubaker, E (w); Cooke, D (a); Allred, M (a) . (2002). Catwoman: The Dark End of the Street. New York: DC Comics.
- Capitainio, A (2010). “The Jekyll and Hyde of the Atomic Age”: The Incredible Hulk as the Ambiguous Embodiment of Nuclear Power#8221;. Journal of Popular Culture 43(2): 249–270, DOI: https://doi.org/10.1111/j.1540-5931.2010.00740.x
- Finger, B (w); Kane, B (a) . (1939). The Case of the Criminal Syndicate!. Detective Comics. New York: DC Comics. (27) Available at: http://www.comixology.com/Detective-Comics-1937-2011-27/digital-comic/10900 [Last accessed 10 December 2013].
- Madrid, M (2009). The Supergirls: Fashion, Feminism, Fantasy, and the History of Comic Book Heroines. United States: Exterminating Angel Press.
- Metz, C (1975). The Imaginary Signifier. Screen 16(2): 14–76, DOI: https://doi.org/10.1093/screen/16.2.14
- Orr, P (1994). The Anoedipal Mythos of Batman and Catwoman. Journal of Popular Culture27(4): 169–182, DOI: https://doi.org/10.1111/j.0022-3840.1994.2704_169.x
- Reichstein, A (1998). Batman—An American Mr. Hyde?. Amerikastudien/America Studies 43(2): 329–350. www.jstor.org/stable/41157373
- Stevenson, R L (1886). The Strange Case of Dr. Jekyll and Mr. Hyde. Seattle: Amazon Digital Services, Inc.. Available at: www.amazon.com/Strange-Case-D...ekyll+and+Hyde [Last accessed 10 December 2013].
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Lesa Syn is an instructor at the University of Colorado in Colorado Springs and Community College of Denver. This article is reprinted from The Comics Grid: Journal of Comics Scholarship.
Catwoman’s Hyde: A Comparative Reading of the 2002 Catwoman Relaunch and Stevenson’s Novella by Lesa Syn is licensed under a Creative Commons Attribution 3.0 International License .


18.5B: Arterial Blood Pressure
The measurement of blood pressure without further specification usually refers to systemic arterial pressure measured at the upper arm.
- Distinguish between arterial blood pressure and venous blood pressure
Key Points
- Systemic blood pressure refers to the pressure exerted on blood vessels in systemic circulation, and is often measured using arterial pressure, or pressure exerted upon arteries during heart contractions.
- Blood pressure (BP), sometimes referred to as arterial blood pressure, is the pressure exerted by circulating blood upon the walls of blood vessels, and is one of the principal vital signs.
- All levels of arterial pressure put mechanical stress on the arterial walls. Higher pressures increase heart workload and progression of unhealthy tissue growth ( atheroma ) that develops within the walls of arteries.
Key Terms
- atheroma : An abnormal fatty deposit that develops within the walls of arteries.
- arterial blood pressure : The pressure of the blood within an arterial vessel, typically the brachial artery in the upper arm. Calculated over a cardiac cycle and determined by the cardiac output (CO), systemic vascular resistance (SVR), and central venous pressure (CVP). It can be approximately determined from measurements of the systolic pressure and the diastolic pressure while there is a normal resting heart rate.
- systemic circulation : The part of blood circulation that carries oxygenated blood away from the heart to the body, and returns deoxygenated blood back to the heart.
The measurement of blood pressure without further specification usually refers to the systemic arterial pressure, defined as the pressure exerted by circulating blood upon the walls of blood vessels. Pressure is typically measured with a blood pressure cuff ( sphygmomanometer ) wrapped around a person’s upper arm, which measures the pressure in the brachial artery. A person’s blood pressure is usually expressed in terms of the systolic pressure over diastolic pressure and is measured in millimeters of mercury (mmHg), for example 140/90.
Blood pressure in the arteries is much higher than in the veins, in part due to receiving blood from the heart after contraction, but also due to their contractile capacity. The tunica media of arteries is thickened compared to veins, with smoother muscle fibers and elastic tissue. Together, these generate of elastic recoil and blood vessel contraction, allowing for the maintenance of a higher pressure.
Typical Tools of Auscultatory Measurement : Shown here are a stethoscope and a sphygmomanometer, used for ascultatory measurement.
Blood Pressure and Cardiovascular Disease
While average values for arterial pressure could be computed for any given population, there is extensive variation from person to person and even from minute to minute for an individual. Additionally, the average arterial pressure of a given population has only a questionable correlation with its general health. However, in a study of 100 human subjects with no known history of hypertension, the average blood pressure of 112/64 mmHg, currently classified as a desirable or “normal” value. Normal values fluctuate through the 24-hour cycle, with the highest readings in the afternoons and lowest readings at night
Changes in Arterial Pressure : Arterial pressures changes across the cardiac cycle.
The risk of cardiovascular disease increases progressively above 115/75 mmHg. In the past, hypertension was only diagnosed if secondary signs of high arterial pressure were present along with a prolonged high systolic pressure reading over several visits. Hypotension is typically diagnosed only if noticeable symptoms are present. Clinical trials demonstrate that people who maintain arterial pressures at the low end of these ranges have much better long-term cardiovascular health. The principal medical debate concerns the aggressiveness and relative value of methods used to lower pressures into this range for those with high blood pressure. Elevations more commonly seen in older people, though often considered normal, are associated with increased morbidity and mortality.
Arterial Hypertension
Arterial hypertension can be an indicator of other problems and may have long-term adverse effects. Sometimes it can be an acute problem, such as a hypertensive emergency. All levels of arterial pressure put mechanical stress on the arterial walls. Higher pressures increase heart workload and progression of unhealthy tissue growth (atheroma) that develops within the walls of arteries. The higher the pressure, the more stress that is present, the more the atheroma tends to progress, and the more heart muscle may thicken, enlarge, and weaken over time.
Persistent hypertension is one of the risk factors for strokes, heart attacks, heart failure, and arterial aneurysms, and is the leading cause of chronic renal failure. Even moderate elevation of arterial pressure leads to shortened life expectancy. At mean arterial pressures 50% or more above average, a person can expect to live no more than a few years unless appropriately treated.
In the past, most attention was paid to diastolic pressure, but now we know that both high systolic pressure and high pulse pressure (the numerical difference between systolic and diastolic pressures) are also risk factors for disease. In some cases, a decrease in excessive diastolic pressure can actually increase risk, probably due to the increased difference between systolic and diastolic pressures. If systolic blood pressure is elevated (>140) with a normal diastolic blood pressure (<90), it is called “isolated systolic hypertension” and may present a health concern.


4.7: Are we really a nation at risk?
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By Ed Farinholt
Learning Objectives
- Identify specific risks outlined in the 1983 "Nation At Risk" document.
- Relate various means that the Public Education system has used to deal with the identified risks
- Name at least one method available to address the qualification risks faced by U.S. Public education teachers
- Identify some choices for increasing teacher motivation from a list of options
Introduction
Having been in school during the Ronal Reagan presidency and coming from a conservative political background these years were exciting to say the least. My friends and family were excited to hear that Dr. William Bennett had been named Secretary of Education and would be investigating the state of education systems in the United States. This article examines several of the risks identified by the National Commission on Excellence in Education headed by Dr. Bennett and assesses their relative merit vis-a-vie the advancements made in the United States public education systems.
Identified Risks
In 1983 the National Commission on Excellence in Education enumerated the following risks faced by the public education systems at that time in an "Open Letter to the American Public" entitled "A Nation At Risk: The Imperative for Educational Reform." (Gardner, 1983) The impetus for this commission, and ultimately this report (hereafter referred to as NAR), was the precipitous decline of the competitive advantage the United States enjoyed in the realm of education compared to that of other countries. (Segal, 2004) While students abroad have long sought advanced degrees from many of the United States noted medical and legal engineering and technological degree programs there has been a noted decrease in the number of Unites States-born students in these schools.
The NAR attempted to grasp for an ultimate direction the United States needed to take to insure the general increase in academic standards for both teachers and students. The directives of this article will focus on the recommendations made toward the public education system since they account for over eighty-seven percent of all United States students. (Hanushek, 1997) While the report from the Commission covers a wide berth of issues pertinent to the state of education, this article will deal with the risks identified in dealing with teachers.
- Teacher qualifications
- Teacher preparation weighted toward methods, vice subject matter expertise
- Salaries are not condusive to attract highly qualified and experienced individuals to the field
- Shortage of teachers qualified to teach such courses as mathematics, science, languages, as well as courses for gifted and talented or special needs students
- Compared to other countries, there are severe deficiencies in the relative competency of both teachers and students in the areas of math and science
Considerations
Note
All, regardless of race or class or economic status, are entitled to a fair chance and to the tools for developing their individual powers of mind and spirit to the utmost. This promise means that all children by virtue of their of their own efforts, competently guided, can hope to attain the mature and informal judgement needed to secure gainful employment, and to manage their own lives, thereby serving not only their own interests but also the progress of society itself. (Gardner, 1983)
While many erstwhile pundits have severely criticized the Commissions findings on the state of education, we will for purposes of this exercise address the ways these identified risks have been addressed to date. In truth, many of these deficiencies have been directly addressed in many of the High School programs throughout the country. We must also remember that public school education monies are largely dependent on local property tax dollars and heavy state and federal subsidies. It could be argued that local business investment in local schools may have a direct impact on supply of qualified graduates.
In practice high schools are designed to prepare students for both degree-oriented academics or vocational careers. In terms of financials, “expenditure outside of instructional staff salaries, going from one-fifth of total current expenditure in 1890 to one-third in 1940 and to more than one-half in 1990. “ (Hanushek, p. 5). School budgets must now cover everything from teachers but extended administrative personnel and counselors with extended social and even psychological skills. During some of my own experience during teacher observation I learned that public schools are now in a situation that all applicants must be accepted and accommodated. In addition, the introduction of technical resources such as computers and sophisticated measurement technologies into the public school classrooms has further extended expenses. When you add the fact that these computers require software, licenses and extensive support resources the price tag for maintenance magnifies each passing year.
Many regard the subsequent reforms imposed upon the education system as "laid down" and often missing the mark with regard to root problems faced by teachers. Teachers must now be equipped to handle a wide array of potential issues in their classrooms, especially with the onslaught of such conditions (properly or improperly diagnosed) as Attention Deficit Hyperac DisorderAcute Hyperactivity Disorder and those of students with English as a second language. While no one ever regarded the costs of providing and maintaining a public education system was ever going to be inexpensive, the jury is still out concerning the return on investment when the public schools attempt to effectively teach under such circumstances. No doubt this is why such programs as Standards of Learning (SOL) and demands for certification of all teachers has been promoted at both the state and federal levels.
Assessments
The publication of the NAR has resulted in numerous studies and controversies attempting to address some of the risks identified. Among those dealing with teacher qualifications, references were made to increased expenditures for both staff and non-staff resources. Schools today are equipped with technical devices such as cable television networks, local area networks, and sophisticated computer software in order to offer students access to the global world of educational opportunities. These in turn require staff who are well versed in these technologies, their maintenance and current uses of them in the global marketplace. The implications of this trend are staggering in terms of the support structure that will be necessary to keep students on an equitable if not a more advanced standing compared to students in other countries.
Teachers are obtaining advanced degrees and in many states must obtain certification from state mandated programs in order to teach. While these have been requirements for a number of years the breadth and extent to which these qualifications cover subject areas pertinent to modern classrooms has significant bearing. Skills such as assessment, familiarity with current technology and multimedia are bearing fruit in terms of involving students and enhancing educator's means of communication and lesson planning.
Tuition tax credits have resulted from the long and bitter battle for vouchers for parents seeking some relief from the taxation burden when in fact they chose to send their children to private schools. Whether for spiritual, advanced placement or special needs, the public school system did not always offer students a suitable environment to excel.
Standards of Learning (SOL) requirements have applied to children in public schools to gage their progress but unfortunately have caused the negative effect of pressure upon teachers to instruct toward specific test goals rather than tailor classes according to the more laudable intrinsic and formative form of pedagogy. The summative aspect of SOLs provide only a benchmark but have treated the objective of learning as the proverbial black box. (Black and William, 1998) Mr. Black and Mr. William contend the focus ought to be on the process of education and the use of formative assessment to insure students are on target with learning objectives and assisted as needed should they fall behind.
Charter, private and home schools have increased in number during the last ten years and bear witness to the increased participation of parents into the educational welfare of their children. Studies have shown that Charter schools have both increased the aptitude of both students who excel in public schools and minority students who would not otherwise have means or access to the advanced technologies or skilled teachers these institutions employ. (Hoxby, 2004) Home schooling have burgeoned and while their effectiveness has largely been at the primary school level, the author's experience is that the issues of limited socialization and technological opportunities faced by home schooled children are more than compensated by innovative parents pooling their collective resources to meet these needs. With this population of students growing yearly it would behoove the public education systems to study their methods.
Another consideration maybe that the problems with education today lay squarely in the laps of parents and guardian involvement in their children's learning. Given that the composition of students throughout the United States has become multicultural, the fact of cultural variances with regard to expectation of parents in regard to their children's knowledge levels brings to bear a much broader element of adaptability by teachers than even ten years ago. The rate of growth of Hispanic full-time teachers increased 2.7 percent to 5 percent between 1998 and 2003. (Weinberg, p. 10) Arguments have been made for bi-lingual education and in many states such as California, Florida and Texas.
Arguments have been made for bi-lingual education and in many area of this country such as Southern California and much of Texas and Florida this may in fact be a necessity. Following an original supposition of this article one could forcefully contend that if answers to educational reform should be indicative of the local community than in fact what happened in the Salinas high school district provides an excellent example of parental activism with regard to not just what their children ought to be taught but how it should be taught. (Arriaza, p. 14) This argument as been argued at a more universal level by Mr. Author Combs who notes that “educational reforms which do not have whole-hearted support of those who must carry them out are a waste of time, effort and taxpayer's money.” (Combs, p. 4)
Another risk which was not broached by NAR was that of morals. When I was in elementary school I remember seeing videos about family planning produced by Planned Parenthool and videos discouraging from drug use like the plague. As an adult, I am offended that I only received part of the story with regard to these ethics while the Christian point of view was not even considered. Today school teachers and administrators cannot even provide an aspirin to a child without a release from the parent or doctor. Yet this same student can receive counseling about such moral decisions as birth control or abuse prevention with no consultation of the child's guardian. (FCPS, 2008)
Conclusion
School reforms ought to be a continual process. Unless public schools are constantly questioning their direction, their curriculum, and their motivations the changing world will continue to force their obsolescence. So in answer to the title of this article, the author proposes a resounding “YES.” Risks are a part of life as everyone can attest. In fact it is the human condition to not only contend with but to face and conquer risks each day. While the right to education by all is firmly upheld in these United States there are many troubles introduced and even some ethical considerations which are inadvertently compromised. The degree to which our nation's educational systems, whether public or private, contend with the ever changing educational requirements will effect the competitiveness and even the moral fortitude of today's students. The ability to effectively teach our young the skills required to think, judge and choose properly, based on sound learning principles, will ultimately decide the degree of success our future leaders will have facing the challenges of the world.
Exercise \(\PageIndex{1}\)
1. A risk identified by the 1983 National Council on Excellence in Education which addresses the ability of teachers to effectively teach physics to High School students would be which of the following findings.
A. Shortage of teachers
B. Teacher preparation
C. Teacher qualifications
D. Teacher salaries
2. County education boards have started to increase the requirements of teacher qualification for freshman and sophomore mathematics teachers through both in-school seminars and mandatory periodic testing addresses which of the following educational reform mandates.\
A. Shortage of teachers
B. Teacher preparation
C. Teacher qualifications
D. Teacher salaries
3. Demands for High School science textbooks to include more realistic lab experiments dealing with such items as water purification and effects of solar power are examples of which of the following educational reform mandates.
A. Industrial currency
B. School Board inattention
C. Staff resources
D. Teacher salaries
4. A vote by a town council in a southwestern Virginia county to allot twenty percent more money for teacher salaries is an example of addressing which of the following reform mandates.
A. Shortage of teachers
B. Teacher preparation
C. Teacher qualifications
D. Teacher salaries
5. Teachers have noticed increasing SOL scores from their students since they have been implementing cooperative efforts to develop formative assessment models for their classes. which of the following recommendations might they make for the enhancements to the state-mandated SOL committees?
A. Include art history related questions
B. Introduce spanish vocabulary questions
C. offer so effective formative questions for inclusion in the SOLs
D. Make the questions addaptive to the level of student's knowledge
- Answer
-
1) B
2) C
3) A
4) D
5) C
References
Arriza, Gilberto, Making Changes that Stay Made: School Reform and Community Involvement , The High School Journal v. 87 no. 4 (April/May 2004), p. 10-24; Retrieved October 2, 2008 from: http://vnweb.hwwilsonweb.com.proxy.lib.odu.edu/hww/results/external_link_maincontentframe.jhtml?_DARGS=/hww/results/results_common.jhtml.29 .
Black, P., & Wiliam, D. Inside the Black Box: Raising Standards Through Classroom Assessment . London: School of Education, King's College. Retrieved October 17, 2008 from: powayusd.sdcoe.k12.ca.us/projects/literacy/SSTTL/AssessDocs/PDFs/BlackBox_Article.pdf.
Combs, Authur W. (1991), The Schools We Need: New Assumptions for Educational Reform . Lanham, MD: University Press of America, Inc.
Gardner, David P.; et al. , (1983), A Nation At Risk: The Imperative For Education Reform , (Washington, DC: U.S. Government Printing Office), Retrieved September 12, 2008 from: eric.ed.gov/ERICDocs/data/ericdocs2sql/content_storage_01/0000019b/80/2f/f7/3d.pdf .
Hanushek, E. A., et al. , Understanding the Twentieth Century Growth in U.S. School Spending ., Journal of Human Resources v. 32 no. 1 (Winter 1997) pp. 35–68; Retrieved September 28, 2008 from ERIC Journal Articles (No. EJ538743).
Hoxby, Caroline M. (2004) Achievement in Charter Schools and Regular Public Schools in the United States: Understanding and Differences . Cambridge, Mass. Harvard University & National Bureau of Economic Research, December 2004. Retrieved October 17, 2008 from: edgeweb.heritage.org/research/education/upload/hoxbycharter_dec2.pdf.
Lewis, Barbar, et al. (2002), Software Compare and Contrast: Blackboard, WebCT and Lotus Notes a Panel Discussion , Informing Science Proceedings. Retrieved October 7, 2007 from http://www.proceedings.informingscience.org/IS2002Proceedings/papers/lewis167softw.pdf .
Segal, Adam (2004) Is America Losing Its Edge? Foreign Affairs , Council on Foreign Relations, November/December 2004; Retrieved October 17, 2008 from: www.foreignaffairs.org/20041101facomment83601-p0/adam-segal/is-america-losing-its-edge.html.
Taylor, Patricia. (2008) Frederick County Public Schools: Student Handbook . Winchester, VA. Retrieved October 17, 2008 from: www.frederick.k12.va.us/education/components/scrapbook.
Weinberg, Sharon L, Monitoring Faculty Diversity: The Need for a More Granular Approach , The Journal of Higher Education (Columbus, OH) v. 79 no.4 (July/August 2008), p. 365-87; Retrieved October 2, 2008 from: http://vnweb.hwwilsonweb.com.proxy.lib.odu.edu/hww/results/getResults.jhtml?_DARGS=/hww/results/results_common.jhtml.20#record_5 .


In short, it is using fancy words used to sound smart or to deliberately confuse your audience. William Lutz called it this inflated language. Most of the time, your audience is confused and not impressed. My dad used to tell me not to confuse my audience or I would be “up the proverbial tributary of deification without and adequate means of propulsion.”

Powerless language consists of words or phrases that weaken the language and undermine credibility. Powerless language results in the speaker being seen as less persuasive, less attractive, and less credible.
It is true that in social settings, you should be willing to use powerless language for the sake of cooperation, but in speeches, you should stick with sounding confident and powerful.
|
Examples of Powerless Language
|
|
Hedges:
Statements that make a phrase sound less forceful.
|
-
I’m
kinda
surprised at the research I found.
-
I
guess
I’d like to…
-
I
think
this point is important because
|
|
Hesitations:
Words or sounds that are pauses in the speech like uh, um, er.
|
-
Uh, I’d like to talk about…
-
W-w-w we can all agree.
-
I wish everyone would, er, uh consider this idea.
|
|
Intensifiers:
Words that do not add meaning but attempt to magnify the emotional content.
|
-
I’m not
very
excited about…
The car is
really
expensive.
I’m
super
excited about this new…
“Substitute ‘damn’ every time you’re inclined to write ‘very’; your editor will delete it and the writing will be just as it should be.” Mark Twain
|
|
Taq questions:
Adding questions to the end of the sentence to make an assertion sound like a question.
|
-
I think this is a great idea,
don’t you
?
It is OK to ask the audience questions. It becomes powerless language when you tag the question on so you don’t sound certain of what you are saying.
|
|
Disclaimers:
Information given before a statement that signals a problem, a lack of understanding, or anticipates doubts
|
-
I’m not sure if I’m saying this right.
-
I probably shouldn’t say this, but…
-
Don’t get me wrong…
-
I know this sounds crazy but…
-
I’m no expert but…
|
|
Self Critical:
Making negative statements about yourself
|
-
Let me start by saying that I’m a terrible public speaker.
-
PowerPoint is not my thing, but here is a slide I made to illustrate.
|
|
Uptalk:
Making voice go up at the end of a sentence making it sound like a question
|
|
Powerless language is not always a bad thing, Dr. Fragale found that when doing group work, powerless language can make you appear more cooperative.
When people hear someone who is very confident and certain in the way that they speak, others think of that person as really dominant and ambitious and assertive, but they also think of that person as less warm, less collaborative and less cooperative. In groups that require a lot of teamwork, team members are looking for people who have good team skills, who care about other people. Those personality attributes are more important than how dominant or ambitious you are.
Oftentimes, you will have a group project that leads up to a speech. In this scenario, you should use your cooperative speech for working with the team and your assertive language in the speech.
At first glance, this whole chapter looks like it is dedicated to things to avoid. In reality, it is dedicated to getting you to think about one big thing–context. Context matters. Who makes up your audience, what the expectations of the occasion are, and who you are in relation to each will impact how you should design your speech. The most important thing in speechmaking is to figure out how to share your message in a way that the audience can listen to and receive your message.
Key Takeaways
Remember This!
-
The goal of your speech is to get your message across to your audience, by knowing the context, the occasion, and the audience you can avoid things that will cause them to not want to listen.
-
Your credibility can be positively or negatively affected by your choice of words.
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Always be intentional with slang, jargon, and big words. Using them or not using them by choice in a way that connects with your audience.
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Always use inclusive language and adapt your vocabulary in a way your audience will feel respected and included.
-
Beware of the curse of knowledge and realize that what is easy for you to understand may not be easy for your audience so adjust your speech accordingly.
Please share your feedback, suggestions, corrections, and ideas
I want to hear from you.
Do you have an activity to include?
Did you notice a typo that I should correct?
Are you planning to use this as a resource and do you want me to know about it?
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Click here to share your feedback.
References
AP. (1992). Perot calls black group ‘you people: Draws fire–some in NAACP audience welcome unpolished speech
. The Seattle Times.
https://archive.seattletimes.com/archive/?date=19920712&slug=1501684
Alda, A. (2017).
If I Understood You, Would I Have This Look on My Face: My Adventures in the Art and Science of Relating and Communicating.
Random House.
Baker, P. (2019). The profanity president: Trump’s four-letter vocabulary.
The New York Times.
https://search.proquest.com/newspapers/profanity-president-trump-s-four-letter/docview/2244183119/se-2?accountid=8361
BBC. (2012). US row over Congressman Todd Akin’s rape remark.
BBC News
.
https://www.bbc.com/news/world-us-canada-19319240
Camerer, C., Loewenstein, G., & Weber, M. (1989). The curse of knowledge in economic settings: An experimental analysis. (1989).
Journal of Political Economy 97
(5).
https://doi.org/10.1086/261651
Cialdini, R.
(1993).
Influence: The psychology of persuasion.
William Morrow.
Dean, H. (2004). The scream that doomed Howard Dean.
https://www.youtube.com/watch?v=l6i-gYRAwM0
Generous, M. A., & Houser, M. L. (2019). “Oh, st! did I just swear in class?”: Using emotional response theory to understand the role of instructor swearing in the college classroom.
Communication Quarterly,
67
(2), 178-198.
https://doi.org/10.1080/01463373.2019.1573200
Generous, M. & Frei, S. & Houser, M. (2014). When an instructor swears in class: Functions and targets of instructor swearing from college students’ retrospective accounts.
Communication Reports. 28
.
https://doi.org/10.1080/08934215.2014.927518
Gura, S. (2010). ‘I’m not a witch,” Republican candidate Christine O’Donnell tells Delaware voters.
https://www.npr.org/sections/thetwo-way/2010/10/05/130353168/-i-m-not-a-witch-republican-senate-candidate-christine-o-donnell-says-in-new-ad?fbclid=IwAR3UsTaVFnHkH4nGk8rmjblbR9ZlbY-3zIKLaydfjdHlQJM0a-mhufvLAno
Haleta, L.L.
(1996) Student perceptions of teachers’ use of language: The effects of powerful and powerless language on impression formation and uncertainty.
Communication Education
45(1), 16-28.
https://doi.org/10.1080/03634529609379029
Harvey, J. (1981). Abillene Paradox. [Video] YouTube
https://www.youtube.com/watch?v=uFQ-ukyhttps://doi.org/10.1080/08934215.2014.927518vAMk
Standard YouTube License.
Heath, C. and Heath, D. (2008).
Made to stick: Why some ideas survive and others die.
New York, Random House
Heath, C & Heath, D. (2006). The curse of knowledge.
Harvard Business Review
https://hbr.org/2006/12/the-curse-of-knowledge.
Hicks, K. (2020). Biden says he regrets ‘you ain’t Black; comment: I shouldn’t have said that”. The Denver Channel.
https://www.thedenverchannel.com/news/election-2020/biden-says-he-regrets-you-aint-black-comment-i-shouldnt-have-said-that
Heyne, R. L. (2016).
“Fired up about education”: A quantitative exploration of positive and negative slang.
Theses and Dissertations
. 572.
https://ir.library.illinoisstate.edu/etd/572
http://doi.org/10.30707/ETD2016.Heyne.R
Hogarth, R. M., & Reder, M.W., eds. (1986). The behavioral foundations of economic theory.
Supplement to Journal of Business, 59
(4).
https://doi.org/10.1177/0739456X8800700214
Hosman, L. A., & Siltanen, S. A. (2011). Hedges, tag questions, message processing, and persuasion.
Journal of Language and Social Psychology, 30(
3), 341–349.
https://doi.org/10.1177/0261927X11407169
Hosman, L. & Siltanen, S. (2006). Powerful and powerless language forms.
Journal of Language and Social Psychology, 25
. 33-46.
https://doi.org/10.1177/0261927X05284477
Jay, T. & Janschewitz, K. (2012). The science of swearing.
Psychological Science.
https://www.psychologicalscience.org/observer/the-science-of-swearing
Jenkins, J. (2002). A sociolinguistically based, empirically researched pronunciation syllabus for English as an international language.
Applied Linguistics, 23
. 83-103. DOI:
10.1093/applin/23.1.83
Johnson
, C. &
Vinson, L
. (1990) Placement and frequency of powerless talk and impression formation.
Communication Quarterly
38:4, 325-333.
https://doi.org/10.1080/01463379009369770
Johnson, C.E.
(1987)
An introduction to powerful and powerless talk in the classroom,
Communication Education,
36(
2),
167-172,
DOI:
10.1080/03634528709378657
Kenan Flagler School of Business (2010). The power of powerless speech.
https://www.kenan-flagler.unc.edu/news/the-power-of-powerless-speech/
Lepki, L. (2020). The internet’ss best list of cliches.
https://prowritingaid.com/art/21/List-of-Clich%C3%A9s.aspx
Lutz, W. (1987). Language, appearance, and reality: Doublespeak in 1984. Institute of General Semantics. ETC 44(4). (no doi)
Mazer, J. P. & Hunt, S. K.
(2008)
“Cool” communication in the classroom: A preliminary examination of student perceptions of instructor use of positive slang,
Qualitative Research Reports in Communication,
9(
1),
20-28,
DOI:
10.1080/17459430802400316
Mazer, J. P., & Hunt, S. K. (2008). The effects of instructor use of positive and negative slang on student motivation, affective learning, and classroom climate. Communication Research Reports, 25, 44 – 55.
doi: 10.1080/08824090701831792
McCusker, C. (2021). Prepone that! Your accent is funny! Readers share their ESL stories.
https://www.npr.org/sections/goatsandsoda/2021/05/16/995963311/prepone-that-your-accent-is-funny-readers-share-their-esl-stories
Pausch, R. (2007). Last lecture: Achieving your childhood dreams.
https://www.youtube.com/watch?v=uFQ-ukyvAMk
PBS Frontline. (2004). Interview with Frank Luntz.
https://www.pbs.org/wgbh/pages/frontline/shows/persuaders/interviews/luntz.html
Reilly, K. (2016). Read Hillary Clinton’s ‘basket of deplorables’ remarks about Donald Trump supporters.
Time Magazine.
https://time.com/4486502/hillary-clinton-basket-of-deplorables-transcript/
Sketchplanations, (n.d.) How to speak plainly by poo.
https://sketchplanations.com/how-to-speak-plainly-by-pooh-bear
Smith, M. (2011). Bafflegab.
bafflegab.net.
Swartzman, E. (n.d.). Why doublespeak is dangerous.
https://www.ericschwartzman.com/why-doublespeak-is-dangerous/
Vinson, L, Johnson, C.E., & Hackman, M.Z. (1993). Explaining the effect of powerless language use on the evaluative listening process: A theory of implicit prototypes. George Fox University Faculty Publications.
https://digitalcommons.georgefox.edu/cgi/viewcontent.cgi?article=1042&context=gfsb
Weiner, T. (2010). Alexander M. Haig Jr dies at 85; was a forceful Aide to 2 Presidents. T
he New York Times.
https://www.nytimes.com/2010/02/21/us/politics/21haig.html


7.4: Two-Dimensional Kinematics with Normal-Tangential Coordinates
Two-dimensional motion (also called planar motion) is any motion in which the objects being analyzed stay in a single plane. When analyzing such motion, we must first decide the type of coordinate system we wish to use. The most common options in engineering are rectangular coordinate systems, normal-tangential coordinate systems, and polar coordinate systems. Any planar motion can potentially be described with any of the three systems, though each choice has potential advantages and disadvantages.
The normal-tangential coordinate system centers on the body in motion. The origin point will be the body itself, meaning that the position of the particle in the normal-tangential coordinate system is always "zero". The tangential direction (\(t\)-direction) is defined as the direction of travel at that moment in time (the direction of the current velocity vector), with the normal direction (\(n\)-direction) being 90 degrees counterclockwise from the \(t\)-direction. The diagram below shows a particle following a curved path with the current normal and tangential directions.
Normal-tangential coordinate systems work best when we are observing motion from the perspective of the body in motion, such as being a passenger in a car or plane. In such cases, we would define ourself as the origin point and "forward" would be the tangential direction. An important distinction between the rectangular coordinate system and the normal-tangential coordinate system is that the axes are not fixed in the normal-tangential coordinate system. If we go back to the car example, the "forward" or tangential direction will turn with the car, but the "east" direction or the \(x\)-direction will remain constant no matter which way the car is pointed.
The way the coordinate system is defined, the position of the particle is always set to be at the origin point. The velocity is also always set to be in the tangential direction, and thus there is no velocity in the \(n\)-direction. The variable \(v\) is the body's current speed.
\begin{align} \text{Position:} \quad &\, r_{p/o}(t) = 0 \ \hat{u}_t + 0 \ \hat{u}_n \\[5pt] \text{Velocity:} \quad &\, v(t) = v \ \hat{u}_t \end{align}
To find the acceleration, we need to take the derivative of the velocity function. This may seem simple, but there is a new thing to consider in that the \(u_t\) unit vector is not constant. This means a change in speed can cause an acceleration, and a change in direction (which would change the \(u_t\) direction) can also cause an acceleration. Going back to our car example, this makes some intuitive sense. We can feel accelerations, and we would be able to feel acceleration if we suddenly stepped on the gas and increased our speed, but we would also be able to feel the acceleration if we took a tight turn at a constant speed.
Going back to the derivative, we will use the product rule, taking the derivative of one piece at a time.
\[ \text{Acceleration:} \quad \, a(t) = \dot{v} \hat{u}_t + v \dot{\hat{u}}_t \]
\(\dot{v}\) is the rate of change of speed of the body, which is called the tangential acceleration. Going back to our car analogy, this is the acceleration we would experience from pressing the gas or brake pedals.
The other piece of our derivative is the speed times the derivative of a unit vector, which we will need to analyze further. When thinking about the derivative of a rotating unit vector, we think about rotating the coordinate system by a small amount \(d \theta\).
The derivative of the \(\hat{u}_t\) vector is then the change in position of the head of the vector. Using some geometry, we can see that the distance the head of the vector moves is the length of the vector (which is always 1 for a unit vector) times the angle of rotation in radians. The direction the head of the \(\hat{u}_t\) vector travels is roughly the \(\hat{u}_n\) direction. In fact, as \(d \theta\) approaches zero, it becomes exactly the \(\hat{u}_n\) direction. Putting this back into our derivative, we wind up with the following equation for acceleration.
\[ a(t) = \dot{v} \hat{u}_t + v \dot{\hat{u}}_t = \dot{v} \hat{u}_t + v \dot{\theta} \hat{u}_n \]
Before we arrive at our final set of equations, we have one last potential substitution. If a particle is moving along a curved path, the rate at which it is turning \((\dot{\theta})\) will be equal to the velocity of the particle divided by the radius of the path at that point (\(v\) divided by rho, \(\rho\)). Putting this last substitution in, we have our final set of equations with two equivalent options for calculating the accelerations.
\[ a(t) = \dot{v} \hat{u}_t + v \dot{\theta} \hat{u}_n = \dot{v} \hat{u}_t + \frac{v^2}{\rho} \hat{u}_n \]
When using these equations, it is important to remember that they are acceleration equations. If we want to know the overall acceleration we would need to add the two acceleration components as vectors . Also, if we are given an acceleration that is not in the normal or tangential direction, we will first need to break that acceleration vector down into normal and tangential components before using the above equations. Finally, if we want the velocity or acceleration in directions other than the normal and tangential directions, we will need to use a coordinate transformation.
A commercial jetliner is traveling at a constant 250 m/s when it executes an emergency 180-degree turn. If the turn takes 20 seconds, what is the acceleration experienced by the passengers? What is the radius of the curve taken by the plane?
- Solution
Assuming a cloverleaf interchange has a radius of curvature of 80 meters at the tightest part of the turn, what is the fastest a car could travel around this curve without experiencing more than half a \(g\) in acceleration? Assume the car is traveling at a constant speed. If the car was instead increasing speed at a rate of 2 m/s 2 , what would be the new overall magnitude of the acceleration experienced by the passengers?
- Solution


19.7: Summary
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The development of technological infrastructure that supports community interactions has seen explosive growth with the emergence of high speed and wireless Internet access. As these tools continue to develop, and tools for integrating disparate information sources become increasingly sophisticated, digital communities have the opportunity to expand and grow. To enhance this growth, greater community involvement in the planning and design of community infrastructure can be used to evaluate the barriers associated with participation. Visual modelling tools are merely one tool that can support this ongoing process of community consultation and development.
Visual models such as the UML notation techniques can help to educate community participants on how information is managed. These tools can also provide important insight into community design initiatives; allowing productive discussions about issues such as privacy, security, intellectual property, and architectural design to occur.
For the greatest community value, it is important to understand how society is being altered by digital technology. Working to inform community discussions can aid participants in understanding digital community design alternatives. These discussions should strive to include real world economic and financial considerations; allowing individuals to invest greater amounts of time and energy into digital communities.
Once individuals begin to invest time and effort into community initiatives, information managers must work to facilitate contributions to community initiatives and develop sequenced candidate plans that support community priorities. The ability to plan and structure digital community development using modelling techniques will allow new ideas to emerge and encourage the coordination and integration of new technological initiatives continually over time.
“The first guiding value in hacker life is passion, that is, some intrinsically interesting pursuit that energizes the hacker and contains joy in its realization”. – Pekka Himanen, The Hacker Ethic, p. 139