miles per hour gives a very different figure to say kilometres per second

This is true in all equations - you must use coherent units, or the numerical answer will be nonsense.

The set of coherent units most used by scientists uses the meter (length), second (time), Kilogram (mass) and other related units such as the Joule (energy) and Volt (voltage).

See:

http://en.wikipedia.org/wiki/International_System_of_UnitsHow can this formula be used in a concrete way to calculate the energy in a given mass of matter?

Actually, the iconic E=mc

^{2} is only part of the equation that calculates the energy in a piece of matter.

The full equation tells us that the energy consists of energy due to the movement of the object, plus a residual energy, even when the object is

*not* moving.

The full equation is this:

E

^{2} = (pc)

^{2} + (m

_{0}c

^{2})

^{2}Where:

- E is the energy in the lump of matter (as measured by you)

- c is the speed of light in a vacuum

- m
_{0} is the mass of the object when it is stationary (not moving, compared to you)

- p is the momentum of the lump of matter (as measured by you)

As we were taught in science class, the kinetic energy of an object increases as it moves faster (compared to us). Surprisingly, Einstein showed that there is an amount of energy (a

*huge *amount of energy) that is present even when the object is stationary (compared to you). This is the mass/energy equivalent.

If you set the velocity=0, then momentum=0, and you are left with the almost familiar E=m

_{0}c

^{2}.

Einstein showed a variety of other surprising things which are also implicit in this equation:

- As the speed of an object changes, it's mass changes.

- There is a maximum speed that a massive object can travel (c)

- The energy of an object when calculated in this way is the same when measured by different observers, traveling at different speeds. (The kinetic energy we calculated at school does
*not* have this useful property.)

- There are things (like photons of light) that have no rest mass, but do have momentum and energy, and they also obey this equation

- If you measure the mass of a moving object (m), and use that in this equation, you do end up with the familiar E=mc
^{2}.

See:

http://en.wikipedia.org/wiki/Energy%E2%80%93momentum_relationhas anyone ever proved it by experiment?

Yes. Every day in summer, the Large Hadron Collider accelerates protons up to an energy of 7 Terra-electron Volts (atomic physicists and astronomers use some non-coherent units to express very small or very large quantities).

If you look up an information sheet on a proton, you will see that it's rest mass m

_{0} is shown as 938 MeV/c

^{2} = 1.672 x 10

^{-27}kg. This means that as the LHC accelerates protons up to almost the speed of light, their mass and energy grows by a factor of about 7,000.

See:

http://en.wikipedia.org/wiki/Proton