The kinetic theory of gases states that temperature is a function of the average kinetic energy of a gas particle. But for the sake of our experiment lets say we have system of 5 molecules each of which are travelling in the same line (positive x direction) with different velocities (5$\frac{m}{s}$,10$\frac{m}{s}$,13$\frac{m}{s}$,43$\frac{m}{s}$,24$\frac{m}{s}$). So to an observer moving in the same direction as the molecules with the velocity (10$\frac{m}{s}$) the molecules in question appear to slowly and hence their velocity is reduced (from the perspective of moving observer). So according to:   Moving observer:  Particles are moving slowly so their kinetic energy and hence temperature (correct me if I am wrong) so these gases would be colder Ground based observer:   Particles are moving fast therefore the temperature of the system is high
In (non-relativistic) classical physics, if the temperature of an object is proportional to the average kinetic energy ${1 \over 2} m\overline {v^{2}}$of its particles (or molecules), then shouldn't that temperature depend on the frame of reference - since $\overline {v^{2}}$ will be different in different frames?  (I.e. In the lab frame  $K_l = {1 \over 2} m\overline {v^{2}} $, but in a frame moving with velocity $u$ relative to the lab frame, $K_u =  {1 \over 2} m \overline {(v+u)^{2}}$).
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