[UPSC IFoS 2018] Thermodynamics (Q65, 15M)

Explain Maxwell-Boltzmann formula for distribution of velocities of gas molecules at temperature T. What will be the formula for distribution of speeds? If \bar{v}, v_{\text{rms}} and v_m denote average speed, root mean square velocity and most probable speed, show that \bar{v} : v_{\text{rms}} : v_m = \sqrt{\frac{8}{\pi}} : \sqrt{3} : \sqrt{2}