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Groups search result 380 for group:rec.running author:Terry author:R. author:McConnell

Search Result 380
From: Terry R. McConnell (
Subject: Re: 10% RULE
Newsgroups: rec.running
View complete thread (8 articles)
Date: 1999/09/12

In article <>,
Ty Young <> wrote:
>It means increase your distance 10% each week if possible, I do this when I
>train for a marathon

Sometimes stated, more generally, that you should not increase milage more than
10% in any given time period.

This suggests a math problem, which, given all the recent bandying about of
mathematics on this group, is timely and even on-topic.

Suppose you take this rule to its logical conclusion: that you cannot
increase your "mileage" more than 10% in ANY time interval. Thus, for example,
the distance you run in the second millisecond after starting your running
program cannot be more than 1.1 times the distance you ran in the first; and
so on. One immediately suspects it would be impossible to even get started, but
this seems to be wrong. 

To formulate a more precise statement, let v(u) be your instantaneous velocity
at time u. Then the rule requires that v satisfy the following:

Integral from t to t+s of v(u)du  <= (1.1)* Integral from t to t+s of

this to hold for all  0 <= s <= t <= T, where T is some finite time horizon.
Since integrals of velocities give distances travelled, this translates into
English as saying that the distance covered in any time interval of  
length s cannot exceed 1.1 times the distance covered in the immediately
preceeding time interval of length s.

Obviously v = 0 is a solution, so a couch potato does not run afoul of the
rule. To make the problem more interesting, one reasons that runners like
to run as far as possible, and so adds the condition: Choose v(u) so as
to maximize Integral from 0 to T of v(u)du  ( total distance covered, )
for v subject to the constraints above. It begins to resemble a problem in
the calculus of variations.

Unfortunately, the problem as stated is still not interesting mathematically
since any constant, but arbitrarily large, velocity is a solution. Such 
solutions require a very ambitious start to the running program, but allow
for no increases in mileage after that. There are also solutions which allow
mileage to increase: the ordinary exponential function, v(u) = e^u, works as
long as T <= ln(1.1). 

Anybody have an idea of a natural additional constraint to add that would 
make the problem more interesting? I suspect at least some initial condition
should be added, or perhaps a bound on the velocity itself.

Terry R. McConnell   Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150                   

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