Forgotten Fundamentals of the Energy Crisis - appendix

by Prof. Al Bartlett


A great deal of correspondence and hundreds of conversations with dozens of people over six years have yielded many ideas, suggestions, and facts which I have incorporated here. I offer my sincere thanks to all who have helped.


When a quantity such as the rate r( t ) of consumption of a resource grows a fixed percent per year, the growth is exponential:

r ( t ) = r0 e k t = r0 2 t / T2         (1)

where r0 is the current rate of consumption at t = 0, e is the base of natural logarithms, k is the fractional growth per year, and t is the time in years. The growing quantity will increase to twice its initial size in the doubling time T2 where:

T2 (yr) = (ln 2) / k » 70 / P         (2)

and where P, the percent growth per year, is 100k. The total consumption of a resource between the present (t = 0) and a future time T is:

C = {T to 0} r(t) dt         (3)

The consumption in a steady period of growth is:

C = r0 {T to 0} e kt dt = ( r0 / k ) ( e kt - 1 )        (4)

If the known size of the resource is R tons, then we can determine the exponential expiration time (EET) by finding the time Te at which the total consumption C is equal to R:

R = ( r0 / k ) ( e kTe - 1 )         (5)

We may solve this for the exponential expiration time Te where:

EET = Te = ( 1 / k ) ln ( k R / r0 + 1 )         (6)

This equation is valid for all positive values of k and for those negative values of k for which the argument of the logarithm is positive.

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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