Just for giggles, when you get down to speeds we deal with, here's what happens.  The current aircraft speed record is held by the SR-71, and is about 3600 km/h, which conveniently enough is 1 km/s.  Rounding off light to be 300,000 km/s, the jet is travelling at 0.00033% of the speed of light.  Plunking that into the Lorentz factor means that the 77 metric tonne Blackbird would, at top speed, have an increased mass of 0.4 milligrams.  What, a flakes of paint, maybe?  That's why we don't notice the effects at the speeds we normally deal with, unless we are being (as in the GPS example) very, very precise.