A Theory Concerning The Expenditure Of Energy On Windfalls
A Theory Concerning The Expenditure Of Energy On Windfalls
"The energy expended in negotiating a system of windfalls results in complete exhaustion of tramper energy (in the limiting case)."
Mathematically, E tends to zero, as T tends to infinity, where E is the energy of the tramper at time T, after the start of the windfall system.
Postulate: Tararua trampers move so fast that the distance between consecutive windfalls is negligible.
The ideal situation is, initial energy, E(o) equals final energy E(f) but of course, if this were so, no energy would be lost, and exhaustion would never occur.
Considering one windfall, there are two fundamental possibilities regarding negotiation of the windfall, each requiring different amounts of energy expenditure.
(a) the windfall may be small enough to step over.
(b) the windfall may not be that small and actual climbing will be necessary to surmount the obstacle.
Experiment has shown that the critical windfall cross-section radius, depends on the knee height and initial energy of the tramper and the slope of the ground.
Since (b) involves larger energies, it will be considered first. It is a known fact that all windfalls are coated with a layer of mud, a very slippery substance. This layer may be thick, as in the case of windfalls that overlie tracks, or thin, (monolayer), when the windfall is fresh. Kinetic energy is lost, due to work done, when climbing up to the top of the windfall and and unsuccessful attempts make this energy loss large. A quantitative expression can be used to describe E(lost).
where n is the number of unsuccessful attempts
m is the mass of the tramper
u is the coefficient of friction between the boots and the windfall (u tends to zero for smooth, muddy windfalls). (note: n and u are fundamentally related).
However the tramper has gained potential energy when he (finally) gets to the top.
E (gain) = mgh
where g is the gravity constant
h is the height of the windfall "summit" above the page 13ground the tramper climbs from.
The tramper turns this potential energy into kinetic energy, by descending to the ground on the other side, but no matter whether the ground is higher or lower on this side, the work done in gaining the top of the windfall is energy lost, the process being irreversible. A rigorous treatment of incremental kinetic and potential energies and work done, shows that this is so and that:-
The sum of all E(gain) is less than E(loss).
This has been experimentally verified for most situations and in fact it has been found that the sum of all E(gain) is approximately zero.
So, E(o) - E(loss) = E(f)
Since E(loss) is greater than zero,
E(o) is greater than E(f)
In case (a), when the windfall is small enough to step over, the energy requirement is less and it can be assumed that the potential energy of the tramper is constant, A fit tramper would barely notice the output of energy required to do work by lifting the knee, and stepping over the windfall, and for a long legged tramper (large knee height), the effect would be small. But this energy requirement implies a negative change in kinetic energy. This process has, of course, a very short lifetime and special techniques would be required to arrive at a quantitative relationship. Suffice it to say, that there is energy loss and again E(o) is greater than E(f). Empirically, E(loss) is proportional to the windfall size, the muddiness (i.e. viscosity) of the ground and the (positive) slope of the ground.
It has been shown, empirically, that for all cases E(o) is greater than E(f) for a single windfall. Remembering the original postulate, it is obvious that a large number of windfalls will drain the tramper's energy. Mathematically, in a crude sense, the E(f) for the first windfall becomes E(o)(1), for the second windfall, which further reduces the energy.
That is, E(o)(i) is much greater than E(f)(j), for i much greater than j, where i and j are the numerical values of two nonadjacent windfalls.
The process continues until E(f)(x) equals zero, when the tramper falls to the ground, exhausted. It is hoped that:
(a) x is large so that a large distance can be covered before exhaustion.
(b) x has a value corresponding to a position coordinate of a hut or other shelter.
x is another (critical) numerical value of a windfall, in a system. Since the value x corresponds to time t tending to infinity, for the average tramper, the final form of the theory is, E tends to zero, as t tends to infinity.
Corollaries:- regarding time for exhaustion.
Several corollaries are obvious, four of which are given;
(1) The more recent the windfall, the quicker will exhaustion occur, since there will be many leafy branches, further impeding progress.
(2) After or during rain, exhaustion will occur sooner since
|(a)||the windfall is more slipperypage 14|
|(b)||the eyes are partially blinded so the optimum angle of approach is not achieved.|
(3) No matter what value the critical windfall radius has, it will decrease as energy is lost, so that exhaustion occurs sooner than that predicted from first principles.
(4) The input of scrog and other fuels, will lengthen the time before exhaustion occurs, but will not lengthen this time infinitely.
The writer was fortunate that he had a large choice of observation sites at which to study, i.e. the Tararuas.
Secondly, he was fortunate in having a good club, the V.U.W.T.C., to assist him in his measurements. (Thanks go also, to the author of the Heels 1963 article on 'Leatherwood Penetration', who put me onto such a helpful club).
There is still much quantitative work to be done on this subject, e.g. what is the value of x? (Here we have an excellent means of grading trampers). It has been observed that x is large for a fit tramper and n is large for an unfit tramper. (n actually increases exponentially as the tramper moves through the windfall system). Most recent Tararua experiments by members of the H.V.T.C., have shown that x is greater than 500 for a F.E. tramper. It is obvious that x is inversely proportional to the mean value of n, but this and the other relationships given in this article, are merely empirical.
All the writer can say, as a word of warning, is that if you must go tramping in windfall country, remember that E tends to zero, as t tends to infinity and therefore camp before t becomes too large.