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No because Relativistic mass the type of mass which they are talking about is not real mass and shouldnt really be called mass
The idea of increasing mass as you get faster relates to kinetic energy that you gain. Who gains the mass when there are two observers involved in an experiment?The one who had his engine running. Physicists are made very aware of relativistic mass increase when they try to accelerate electrons to moderately high velocities. If you use a simple cyclotron, then they just reach a limit to their top speed. You need to modify the magnetic field or driving frequency to get higher speeds to make the increased masses go in a circle of appropriate diameter. It's called a betatron.I'm not sure how that relates to your foregoing statements, lightarrow. Of course, the electrons aren't in an inertial frame when they're in the betatron.
Even in linear motion, the energy for incremental speed change gets disproportionately high as you approach c. The simple m v squared /2 no longer applies. Why does this not imply an increase in mass?