First order relativistic correction to Newtonian kinetic energypic.twitter.com/1dFEVb4bG1
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“Small” is always in comparison to something else. If one needs extreme precision, as for example in measuring the “fine” spectrum of hydrogen, even this “small” relativistic correction plays a major role.pic.twitter.com/CHt7iNfBh5
Yes of course, the equation for relative velocities would be significant at near light speeds but that would require vast amounts of energy. 

Not necessarily .. consider neutrinos
Just making a friendly observation, that's all.
Small velocities n large masses ~ no effect Large velocities n negligible mass ~ no effect Large velocity n non-negligible mass - this correction is significant. Then such masses will have large energy too, so corrections should be instantaneous?(otherwise need big acceleration)
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