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...because as its velocity increases so does the mass... (blah-blah-blah - I'm not going to explain the whole argument in detail because you probably you know how it goes) ...therefore it would take an infinite amount of energy to reach the speed of light.
It is true that for a stationary observer the mass of a particle (from now on it's going to be a spaceship, because they are awesome) increases as it accelerates.
Observers at rest with respect to the rocket ship (or particle) will measure it to have the same mass it had before it started to move.
But in the frame of reference of the ship (as seen by, for example, passengers on board) the mass of the ship is equal to its rest mass - no relativistic mass is added. In fact, for them, the opposite happens - the mass of a particle, they were stationary to before they started moving (from now on it will be called "the Earth"), is now increasing.
in the frame of reference of the ship, it's of the same mass as when it started
For t < 0 a rocket ship and a particle is at rest in the inertial frame of reference S. When t = 0 the rocket ship starts to accelerate while the particle remains at rest. For observers at rest in frame S the particle's mass remains constant having the value of its proper mass. The mass of the rocket as observed from this frame is increasing. From the rocket's frame of reference S' the rocket's proper mass remains constant (disregarding fuel ejected to create thrust. So by "rocket" we'll mean "command capsule" where astronauts live) but observers in S' will determine that the mass of the rocket is constant but that the particle's mass is increasing.
For more on this please see my website on this at: http://home.comcast.net/~peter.m.brown/gr/uniform_force.htm
However, from the viewpoint of the observers on the ship, light still passes them at speed c, so despite burning lots of fuel, they have made no progress towards their goal of traveling at speed c. So they will need infinitely more fuel than they have already burned to reach speed c.
So is it possible for two objects to travel at/faster than c relative to each other within the observable universe?
So is it possible for two objects to travel at/faster than c relative to each other ...