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Where do these constants in equations come from?

  1. Nov 30, 2007 #1
    Something I've always wondered: how did Physicists and Mathematicians of years past discover equations with these integer (and even fraction) constants in them?

    Take for example the mean-square-speed equation:
    [tex]\mu \equiv \sqrt{\frac{3RT}{M_{m}}}[/tex]

    or Kinetic Energy:
    How do they discover this 3 and .5? It seems arbitrary.
  2. jcsd
  3. Nov 30, 2007 #2


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    This 3 comes from dimensionality of space... which plays a role in the statistical treatment of the ideal gas law.
    This .5 comes from the integral of v dv, which arises from the definition of work and Newton's Second Law.

    These relations are derived from first principles...
    Last edited: Nov 30, 2007
  4. Nov 30, 2007 #3


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    The integer and even fraction constants are usually pretty apparent when you go through a derivation.
  5. Nov 30, 2007 #4


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    Hyperphysics is one good site to explore some of these questions, e.g.

  6. Dec 1, 2007 #5
    Thanks for the links Astronuc.

    Something I forgot to leave out of my original post is my surprise at how often it turns out to be an integer (or simple fraction) that it is multiplied by.
    Although I guess going through and deriving it shows why.
  7. Dec 1, 2007 #6
    When it comes to empirical investigation of constants, remember that, say, 100000000000000000000/100000000000000000001, is pretty close to 1. It is convenience to choose one; and the fact that there is no measurable difference.
  8. Dec 2, 2007 #7
    its not ALWAYS an integer... A famous number for example is the golden ratio, which is like (1+sqr(5))/2 ~= 1.62
    Or... number pi? Area of Circle = pi*r^2 ?
    Many times in formulas you have square roots of things, which are irrational numbers... there are PLENTY of examples where the constants are not integers.
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