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Originally Posted by Lil' Lavery
"Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same velocity."
Well, if you consider the position equation f(t)=g(t)-h(t) (g being the position of one runner and h the other). Then the velocity equation would be f'(t)=g'(t)-h'(t), so if the velocities are the same, f'(t)=0. When t=0 and when the race end (another value t), and possibly other ponts, f(t)=0.
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Everything about your argument is fine up to this point. f'(t) must be zero for some t between the start and finish of the race. This can be proved using the Mean Value Theorem; the result is called
Rolle's Theorem.
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And because the derivative of a constant is 0, then f'(t) would also be 0.
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This does not follow from the problem statement. In practice a runner's velocity at the finish line is rarely zero, and in some kinds of racing (relays, sailing, etc.) the starting velocity is not zero, either. But on a purely mathematical level, the statement "the derivative of a constant is zero" is neither precise nor pertinent here.
Precisely, the derivative of the function f(t) = C is zero for all t. However, in this example you have assumed that the function f(t) = 0 at the beginning and the end of the race, but not for times before the beginning or after the end. What the two runners did before t=0 and after crossing the finish line is not defined, nor is that information needed to prove the result.
__________________
Richard Wallace
Mentor since 2011 for FRC 3620 Average Joes (St. Joseph, Michigan)
Mentor 2002-10 for FRC 931 Perpetual Chaos (St. Louis, Missouri)
since 2003
I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
(Cosmic Religion : With Other Opinions and Aphorisms (1931) by Albert Einstein, p. 97)