This is a very important expression, so we should check that, at low speeds, it corresponds to the classical expression. Using the expansion valid for small v/c: so as shown in the graph at right - the purple line is ½mv2+mc2. Ah, that's reassuring. But it also shows us Einstein's speed limit. In the classical expression, v is proportional to the square root of K. In principle, v could be as large as one liked. In relativistic mechanics, however, (γ - 1) increases linearly with the kinetic energy. And γ goes to infinity as v goes to c. A finite amount of energy can never accelerate a body to the speed of light. So now let's look at equation (*) again: Where did the 1 on the right hand side come from? It's the starting value of the integral. Now all of the terms in this equation are energies. When γ > 1, we have non-zero kinetic energy. So, if we think of γmc2 as the total energy of body, and write then (γ - 1)*mc2 is the kinetic energy, and 1*mc2 is an energy that a body has when v = 0 and γ = 1. Remember the proper time and proper length of module 4. These were the time and length of a body measured in its own frame. So we could write where E0 = mc2 is the proper energy of a body - the energy that it has, even when it is not moving. We are getting used to large numbers, but let's remember that c2 = 9.0 x 1016 m2s-2 = 9.0 x 1016 J/kg. Yes, c2 has units of joules per kilogram, as we can see from the familiar expression Kclassical = ½mv2. So the "destruction" of one kilogram would liberate nearly 1017 joules. Ouch! This is both the promise of nuclear energy supply, and the terrible fact of nuclear weapons. (See also the discussion of binding energy.)

A moderately large power station might provide power at 1 GW or 10⁹ joules/second. To provide this power continuously for a year requires 3 x 10¹⁶ joules. So, in such a power station, about 300 grams of matter would disappear each year. In a nuclear power station, the fuel rods would weigh less when you took them out than when you put them in. (Because this is a small amount, the actual accounting is complicated by a range of processes that occur to the rods.)

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.

As another example, consider a bomb that reduces the total mass of its fuel. Suppose that the blast from this bomb pushes the atmosphere out of the way in a hemisphere of radius R. (It is said that, in the Manhattan project, one of the characters uses the displacement of pieces of debris to estimate the yield after the first test. A simple calculation that one can do in one's head gives the yield:

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.