This is where I've recorded my own exploration of this topic:
There is a little robotic space probe floating out in space that is not accelerating. We can take it's inertial frame of reference as v=0.
The engine, when on, provides 1 watt of power which is converted to kinetic energy. The probe has a mass of 2 kg. After one second, the probe has 1 J of kinetic energy, and it's velocity is
1 J = (2 kg)(v^2)/2
1 J/kg = v^2
1 (m/s)^2 = v^2
1 m/s = v
OK, so after one watt is used to accelerate the probe, it is now moving at 1 m/s with respect to the orginal frame of reference.
Now, I want to present two chains of reasoning:
A) If I run the engine for three more seconds, the probe will move at a speed of 2 m/s with respect to the original frame of reference as calculated by K=mv^2/2.
B) According to classical relativity, all inertial frames of reference are equivalent in terms of how the laws of physics acts within them. Thus, after the first second, we can choose a new inertial frame of reference that moves with the ship. Now the ship has v=0 and K=0 with respect to the new frame of reference.
We let the engine run for 1 s. Now the ship has a velocity of 1 m/s with respect to the new frame of reference. If we reset the frame of reference every second, this process will continue each second.
After four seconds, the probe is moving at 1 m/s with respect to the fourth frame of reference. The fourth frame of reference is moving at 1 m/s with respect to the third frame of reference, so the probe is mobing at 2 m/s with respect to the third frame...and 3 m/s with respect to the second frame...and the probe has a velocity of 4 m/s with respect to the probe's original velocity four seconds ago.
Why am I getting different answers?
2/9/10 A person whom I shall call BrotherGeeWhiz commented on my confusion:
I think the issue would be resolved if you took into account the acceleration, which is the quantity I like to think about in classical mechanics. It's proportional to force and invariant to changes in inertial reference frames. If one proposes some continuous form for the acceleration, then the velocities should add along different reference frames (as long as v < < c).
Also (tiny point), for an isolated mass in space to move, it has to propel some "fuel" in the opposite direction to conserve momentum. This changes the mass, but since the fuel can move arbitrarily fast, the change in mass can be made arbitrarily small.