Relativity in brief... or in detail..
Energy in Newtonian mechanics and in relativityUnder relativity, the laws of physics may be the same for two observers with relative motion, but they sometimes look unfamiliar to those of use who are used to putting kinetic energy = ½mv2. In this animation, a rocket engine does work at a constant rate, ie it produces constant power. (This is not the usual behaviour or rockets, but it means that we can see what increasing the kinetic energy does to speed and momentum under classical and relativistic mechanics. We discuss this animation further below.
Another way of writing Newton's second law in classical mechanics is the work energy theorem. By integrating the resultant force F that acts on a body with respect to the distance over which the centre of mass moves, one obtains the important result that the work done by F equals the change in the quantity ½mv2, a quantity that we call the kinetic energy. As you might expect, things are a bit more complicated in relativity. Let's start with Newton's second law, that states the force is equal to the time rate of change in momentum:
In a coal fired power station with the same continuous power output, 300 grams of matter would also disappear. Yes, if you could gather all the CO2, H2O, soot, nitrides and other products of the power station (ah, if only we could gather them!) then we should have 300 grams less than the coal and air that we reacted in the furnaces.
Compared to the coal fired station, the nuclear station has several advantages. It makes little contribution to global warming. Its waste products are small and well localised, which is an advantage when collecting storing them. And its mining operations are relatively safe compared to coal. On the other hand, many people don't like living near nuclear stations whereas they are less perturbed by the proximity of coal-fired stations. Further, it is sometimes argued that enemies might use the nuclear station as a target, so such stations need to be build with that in mind.
The work done against the atmosphere is atmospheric pressure times 2πR3/3, the volume of the hemisphere. Suppose that R is 100 m, substitution shows that this requires the loss of 2 micrograms of matter. Again, it doesn't matter whether it is a chemical or nuclear bomb. Fortunately, extremely large chemical bombs are fortunately hard to deliver. I regret to say that nuclear bombs small enough to be carried in a backpack or fired from a cannon, but with huge destructive power, have been developed by several countries.
Notice Einstein's speed limit: the Space Ship Newton can accelerate indefinitely, but the SS Einstein approaches c as the energy put into it goes to infinity. On the other hand, the momentum rises more rapidly with input energy under relativity.
For more about Newtonian and Galilean mechanics, see Physclips: Mechanics with animations and film clips.