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$E=\frac 12mv^2 => v=\sqrt{\frac{2E}{m}} $ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count. An object may vibrate or rotate even if it's centre of mass has zero speed.

As each available degree of freedom has the mean energy $E=\frac 12kT$, and as there are 3 independent translational degrees of freedom, the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.

$E=\frac 12mv^2 \implies v=\sqrt{\frac{2E}{m}} $ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count. An object may vibrate or rotate even if it's centre of mass has zero speed.

As each available degree of freedom has the mean energy $E=\frac 12kT$, and as there are 3 independent translational degrees of freedom, the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.

$E=\frac 12mv^2 => v=\sqrt{\frac{2E}{m}} $ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count. An object may vibrate or rotate even if it's centre of mass has zero speed.

As each translational degree of freedom has the mean energy $E=\frac 12kT$,
  the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.

$E=\frac 12mv^2 => v=\sqrt{\frac{2E}{m}} $ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count. An object may vibrate or rotate even if it's centre of mass has zero speed.

As each available degree of freedom has the mean energy $E=\frac 12kT$, and as there are 3 independent translational degrees of freedom, the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.

$E=\frac 12mv^2$ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count, as an object may vibrate or rotate even if it's centre of mass has zero speed.

As each translational degree of freedom has the mean energy $E=\frac 12kT$,
the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.

$E=\frac 12mv^2 => v=\sqrt{\frac{2E}{m}} $ is valid for translational kinetic energy and the speed of the centre of mass.

Vibrational or rotational energy does not count. An object may vibrate or rotate even if it's centre of mass has zero speed.

As each translational degree of freedom has the mean energy $E=\frac 12kT$,
the mean translational kinetic energy is $E_\mathrm{transl}=\frac 32kT \le E_\mathrm{kin}$

and the respective $v_\mathrm{RMS}=\sqrt{\frac{2E_{transl}}{m}}= \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$

even for ideal gas consisting of multiatom molecules.