Where do these constants in equations come from?

[SticksandStones]

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:
$$\mu \equiv \sqrt{\frac{3RT}{M_{m}}}$$

or Kinetic Energy:
$$\frac{1}{2}mv^{2}$$
How do they discover this 3 and .5? It seems arbitrary.

 

[robphy]

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:
$$\mu \equiv \sqrt{\frac{3RT}{M_{m}}}$$

or Kinetic Energy:
$$\frac{1}{2}mv^{2}$$
How do they discover this 3 and .5? It seems arbitrary.

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...

 

[Mk]

The integer and even fraction constants are usually pretty apparent when you go through a derivation.

 

[Astronuc]

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:
$$\mu \equiv \sqrt{\frac{3RT}{M_{m}}}$$

or Kinetic Energy:
$$\frac{1}{2}mv^{2}$$
How do they discover this 3 and .5? It seems arbitrary.

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

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/kintem.html
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/molke.html
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/maxspe.html

 

[SticksandStones]

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.

 

[Moridin]

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.

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.

 

[Mephisto]

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.