Imagine two unit mass balls with a compressed and latched spring between them. Unlatching the spring imparts a velocity of one to each ball. Lets call the energy of a unit mass ball moving at unit velocity one. The spring must store two energy units as it is able to impart one unit to each ball. Now imagine that we start the two balls moving in the same direction with unit velocity before we unlatch the spring. The velocity is along the axis connecting the centers of the balls. The kinetic energy of the balls is one unit each, or two. The spring also stores two units. We now unlatch the spring and one ball stops dead while the other proceeds with a velocity of two. All of the system energy is now in the second ball. The total energy is four. Thus a unit ball moving at velocity two must have four times the energy of a ball moving with velocity one. The factor of one half is another less interesting story.
It is not immediately clear what concept will serve for energy when the geometry of space time is as specified by Einstein. We will keep conservation in the sense of the above experiment but find a modified formula involving mass and velocity. The formula must be of the form E = m·f(v) directly from conservation. Since a velocity of 1 is special in relativity we recast the above to imagine that the ball’s velocity after starting from rest is 1/2 and the velocity imparted to the coupled pair is also 1/2. Also we take the velocity of a particle that moves without acceleration from <x, t> = <0, 0> to <1/2, 1> to be 1/2. (Rapidity = β is another candidate to inherit the name ‘velocity’ in the new world! tanh(β) = v.)
We posit that the spring can impart velocity 1/2 to each ball. If first we start the two balls with velocity 1/2, then unlatch the spring, we compute the velocity of the 2nd ball by transforming into the frame where the latched balls are not moving. …
I pause here in mid thought to report that Penrose on page 434 of The Road to Reality addresses just this point. There he raises and answers important issues.