Forgotten Fundamentals of the Energy Crisis - Part 2

by Prof. Al Bartlett

II. Background

When a quantity such as the rate of consumption of a resource (measured in tons per year or in barrels per year) is growing at a fixed percent per year, the growth is said to be exponential. The important property of the growth is that the time required for the growing quantity to increase its size by a fixed fraction is constant. For example, a growth of 5% (a fixed fraction) per year (a constant time interval) is exponential. It follows that a constant time will be required for the growing quantity to double its size (increase by 100 %). This time is called the doubling time T2 , and it is related to P, the percent growth per unit time by a very simple relation that should be a central part of the educational repertoire of every American.

T2 = 70 / P

As an example, a growth rate of 5% / yr will result in the doubling of the size of the growing quantity in a time T2 = 70 / 5 = 14 yr. In two doubling times (28 yr) the growing quantity will double twice (quadruple) in size. In three doubling times its size will increase eightfold (23 = 8); in four doubling times it will increase sixteenfold (24 = 16); etc. It is natural then to talk of growth in terms of powers of 2.

Reprinted with permission from Bartlett, A., American Journal of Physics, 46(9), 876, 1978. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Association of Physics Teachers. Copyright 1978, the American Association of Physics Teachers.
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