And how is the earth moving at 0 m/s? It is moving at several kilometres a second through space.
Relative to what? The sun? Well surely the sun isn't stationary; it is moving through the galaxy. Relative to what? The center of the galaxy? Well surely the center of the galaxy isn't stationary; it is moving through the universe. Relative to what? The center of the universe? Well surely that isn't stationary, it's got to be moving in relation to something.
You can pick all the arbitrary frames of reference you want, but
to us, and all the things on earth, the earth isn't moving. The velocity of earth (relative to me) is defined as V(earth absolute) - V(me absolute). Ignoring the fact that there is no such thing as absolute velocity, that fact remains that the earth and I are moving at the same speed, so V(earth relative to me) will always be zero.
If I throw a baseball at 20 m/s, then the velocity of the earth is 20 m/s relative to the baseball. At 20 m/s, we aren't going to be experiencing a noticeable increase in mass or anything.
If I throw the baseball at (.8)C, then we are moving at (.8)C relative to the baseball. The baseball is going to think we gained a hell of a lot of mass, because (.8)C is well into the "crazy physics" region.
To me, however, the earth is going just as fast as it was before I threw the baseball; that is to say, it's not moving at all. Since the Earth isn't moving at all for me, it won't have gained any mass for me.
The baseball, however, will look pretty damn heavy.