Reading: none

Movie- Forgotten Fundamentals of the Energy Crisis , by AA Bartlett.

• "Absolutely riveting!," Rosie (the) Riveter, Steel Structure Journal.
• "A fascinating juxtaposition of concepts and graphics with, well... ideas and graphs," Sir Oliver Stone-Lb, Royalty Weekly.
• "An old, male curmudgeon yelling at us about exponential growth and the energy crisis-- a lifelike triumph of reality and algebra over style and ignorance," Prof. Livelybrooks, The Extremely August Journal of Physics and Movie Review.

Exponential Growth-- a Review.

Movie- Forgotten Fundamentals of the Energy Crisis , by A.A. Bartlett.

Dr. Bartlett may be curious to look at, but he's absolutely correct in his analysis. Bear with him as he helps us examine the concept of exponential growth and its many implications for society, including:

• Population growth
• Oil production and consumption
• The use of coal in the U.S.
• What the experts say when they aren't thinking clearly (clearly aren't thinking?).

Exponential Growth-- a Review.

Traveling to Portland, an example:

Imagine that we traveled at a constant speed (100 km/hr) to Portland from Eugene (150km). We would arive there in 1.5 hours. We know this because:

distance covered = speed x time elapsed, or

Dd = s D t

In this instance the speed is constant. It doesn't care about how far or how long we have traveled. It is the same in Salem as it was in Albany.

A graph of the relationship between distance and time for a constant speed would look like the blue dashed line, below:

What if we traveled to Portland such that our speed is a function of the distance traveled? More specifically, what if our speed increases according to how far we have traveled? We would write this proposed relationship as:

speed = lambda x distance, or

s = l d

Where l is some constant. Given our relationship between speed, distance and time, we can rewrite this as:

speed = distance covered / time elapsed

= lambda x distance, or

s = Dd / D t = l d

Now, in driving to Portland, our speed increases according to how far we have traveled.

The slope of our distance-time graph, therefore, must increase as time (and, hence, distance) increases. In fact, by the time we reach Portland we are traveling at about 470 km/hour!

{The general solution to this equation can be found using the methods of calculus. We don't do calculus (or windows) so I'll state the solution:

d_o is the initial distance at time t=0, and "e" represents a special number (approx. 2.7183).}

This type of equation, the exponential relationship, correctly predicts the behavior over time of a many biological and natural phenomenon. For example, the relationship between "change in population" (per unit time interval) and population, p, is given by:

Important bit in words:

The change in population over a given time interval (say a year) is proportional to the population itself.

The more people there are, the more babies one might expect.

We can use the above equation, then, to predict the population (p) at a later time (t), given the popluation now (p_o) and given the growth rate, l. As a consequence, population vs. time curves look like our graph of exponential growth, the pink, solid line above.

Doubling time:

One can also solve for the doubling time, the time it takes for the population, for example, to become twice as large as before. {This happens when l t = 0.693 (e^0.693 = 2).} In practical terms, the doubling time can be determined using the expression:

The % growth per year is related to l. The larger l and, hence, the % growth, the smaller the doubling time. Consequently it requires less less time to double the population for a larger l.

• One other important thing to remember about exponential growth. The amount used during a doubling period is equal to the total amount used before that period.

Logarithmic graphs

One last word about graphs and exponential growth. One can plot our speed in traveling to Portland exponentially, for example, in a special way by setting the vertical graph axis to represent logarithm of speed. A logarithm is a special mathematical function that solves the exponential growth equation. An example semi-logarithmic graph depicting this is shown below.

Note that now the graph is a straight line. It still depicts an exponentially-changing relationship between distance and time, because each numbered jump in the vertical axis represents a change by a factor of ten (multiplication by 10).

So one must be careful when reading graphs.

The relationship depicted above still predicts a speed of 470 km/hour upon reaching Portland, and speed (or population growth) is still increasing over time. For that matter, distance (or the number of people on Earth) also increases more rapidly than for a linear relationship.

Moral: Be careful when reading graphs. You could be deceived into thinking that things are really OK....

Coming soon to a theater near you...

In the next lecture we will see how Dr. Bartlett's scenario has played out 17 years later. We will be examining US historic energy consumption and looking at what could lie ahead. We will focus specifically on the consumption of and prospects for fossil fuels.

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