1) Kinetic energy has been measured as KE=½mv²
2) The speed of light has been measured as the same regardless of the speed of observers.
If an omnidirectional light source in empty space emits light for 1 hour, it will reduce its total mass/energy as it emits light. If the light source then accelerates away, it will gain KE=½mv².
-E+KE2
If the light source accelerates away without emitting light, it will gain KE=½mv² based on its original mass. If the light source then emits light for 1 hour, this will reduce its mass/energy by the same -E as in the first case. However, the static observer will measure the amount of light emitted as E(1+v²/2c²) Light travels at the same speed for all observers, but it appears red shifted if light sources are moving away from the observer.
KE1-E(1+v²/2c²)
However, in both cases, the total light energy plus KE is the same.
-E+KE2=KE1-E(1+v²/2c²)
E(v²/2c²)+KE2=KE1
KE=½mv² Since the velocity is known to be the same in this case, the mass must have changed.
E(v²/2c²)+½m2v² =½m1v²
E/c²+m2=m1
E/c²=m1-m2=change of mass
E=mc²