KE=1/2mv^2 refers to a single particle - so this will give a specific velocity of a single particle
KE = 1/2mv(rms)^2 is average velocity of a population of particles - this is used in conjunction with equipartition theorem (which pretty much says that the avg. kinetic energy of all particles in a population is approximately equal in all translational directions - 1/2Kb*T in one direction, so 3 times that for all 3 dimensions); i.e. 1/2mv(rms)^2=3/2*Kb*T
When you solve this equality you'll get the "Urms" eqn you posted above. Doing this requires a fundamental phys chem relationship: R = N*Kb; here N is Avogadro's number (i.e. number of particles per mole).
Derivation:
1/2mv(rms)^2=3/2*Kb*T --> v(rms)^2 = 3KbT/m
Hence, Kb = R/N --> v(rms)^2 = 3(R/N)T/m = 3RT/Nm
m = mass of single particle; hence Nm = mass of a mole of particles --> i.e. molecular mass (which you designated as 'M')
Hence: v(rms) = sqrt(3RT/Nm) = sqrt(3RT/M)
In short, v is velocity of a particle; v(rms) is average velocity in a population of particles.
The "most like" velocity you refer to is I'm guessing the most probably velocity which is DIFFERENT from the v and v(rms). This is found by calculating the maximum of the Boltzmann distribution; in a standard Gaussian bell curve, you'd expect average and most probable to be identical b/c a Gaussian curve is symmetric; because Boltzmann distribution is asymmetric, they are not the same - v(rms) always greater than v(probable)
