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One joule is required to accelerate a 1 kg mass from zero to one meter per second per second.

ie 1m/s2.

Now, to further accelerate the mass to 2 meters per second requires an additional 3 joules... because at that velocity, the energy now possessed by the mass is 4 joules.

Ok, let's say I use 1 joule of energy to accelerate the mass to 1m/sec.

Now, say we have train passing by with velocity 1m/sec and going in the same direction as the mass. A person on that train grabs hold of the mass. Because it's and the trains speed are equal, the mass is effectively at rest from the train's frame of reference.

Now let that person - on the moving train - apply 1 joule of energy to again accelerate the mass to 1 m/sec - but relative to HIM this time. So the actual velocity of the mass is now 2m/sec, relative to the first observer.

So only TWO joules of energy have been required to get the mass to 2 m/sec instead of the FOUR implied by the KE equation.

I have no difficulty in grasping that energy goes up as the square of velocity, but for the life of me, I cannot understand why both observers cannot use 1 joule of energy to effect an increase of 1 m/sec each

Where am I getting stuffed up here ?