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One equals Two

Date: 07/25/2001 at 17:36:12
From: Dick
Subject: One equals Two

There is an algebraic manipulation involving division by zero that 
results in one equals two, or some other contradiction. What is it?

This is a physics "proof" that one equals two.

Proof That One Equals Two

Beginning with the well known equations for uniform motion:
    s = 1/2(a)t^2
and solving each for acceleration, a :
    a = v/t and
    a = 2(s)/t^2
and setting them equal to each other:
    v/t = 2(s)/t^2
And using the fact that s = vt and substituting:
    v/t = 2(v/t)
Thus factoring each side by v/t:
    1 = 2
Ta Da!

Date: 07/25/2001 at 22:56:23
From: Doctor Peterson
Subject: Re: One equals Two

Hi, Dick.

We have a FAQ that gives several of these false proofs:

   False Proofs, Classic Fallacies   

You are probably aware that the main use of such fallacies is to teach 
us the dangers of unthinking manipulation of equations. That is just 
as important in physics as in math.

In your physics "proof," the problem is that you are using formulas 
that apply to different situations. The first two apply to the same 
falling body, at a fixed time t after being dropped (or exposed to a 
force), which results in velocity increasing from zero. But s = vt 
applies to a constant velocity, not a constant acceleration. And in 
fact, under constant acceleration, the average velocity happens to be 
just half of the instantaneous velocity, which accounts for the ratio 
of 1:2 you found.

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Number Theory
High School Physics/Chemistry

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