Changes

The equation is about energy, both kinetic energy and potential energy.

The equation is about energy, both kinetic energy and potential energy.

Kinetic energy is actual visible energy, that is, energy of things that are in motion.  (Or light or other electromagnetic radiation; this is the motion of photons.  Light conveys both energy ad momentum.)

Kinetic energy is actual visible energy, that is, energy of things that are in motion.  It's the energy of a thrown baseball.  Radiation (for example, light) also counts as kinetic energy—it's the motion of photons.  Light carries energy, force, and momentum. The force carried by light is not as obvious as the force of a thrown baseball, but it is there.  The force of sunlight has been proposed for long-term space travel.  It is also the force that causes the [[Pioneer anomaly]] and the force that makes a comet's tail stream away from the Sun.

Potential energy is "hidden" energy that can become kinetic energy, or vice-versa.  A wound up spring, a charged battery, a stretched rubber band, a mixture of gasoline and air, an explosive, and a radioactive atom, all have potential energy.

Potential energy is the other kind—"hidden" energy.  It can become kinetic energy, or vice-versa.  A wound up spring, a charged battery, a stretched rubber band, a mixture of gasoline and air, an explosive, and a radioactive atom, all have potential energy.  It's what is needed to make the principle of conservation of energy work.  That is, when kinetic energy comes into existence, it's because potential energy was converted into kinetic energy.  The wound-up spring of a clock has potential energy, that runs down over time, being converted into the kinetic energy of the ticking sound.  A battery has potential energy that runs down when it provides electricity to make things move.  Various chemical substances have characteristic amounts of potential energy, that may be converted to or from kinetic energy when chemical reactions occur.  For example, Sodium and Chlorine have more potential energy than Sodium Chloride.  Explosives have more potential energy than their constituent atoms.  Radioactive atoms have more potential energy than their "daughter" atoms.

The principle of ''conservation of energy'', universally accepted for well over 100 years, says

The principle of ''conservation of energy'', universally accepted for well over 100 years, says

::Total energy (kinetic + potential) is always conserved.

::Total energy (kinetic + potential) is always conserved.

Hundreds of years of research by chemists (and, before that, the alchemists) worked out the potential energies that are characteristic of various substances, and that the potential and kinetic energies are accurately converted from one to the other, leading to the principle of conservation of total energy.

Hundreds of years of research by chemists (and, before that, the alchemists) worked out the potential energies that are characteristic of various substances, and that the potential and kinetic energies are accurately converted from one to the other, leading to the principle of conservation of total energy.

::'''Potential energy has mass.'''

::'''Potential energy has mass.'''

That is, it weighs something.  Whenever anything has potential energy of any kind in it, improbable as this may sound, it weighs more.  The proportionality constant is 1/c², or 1.11 x 10^-17 kilograms per Joule.  A fresh battery weighs more than a spent one, a wound-up alarm clock weight more than a run-down one, etc.

That is, it weighs something.  Whenever anything has potential energy of any kind in it, improbable as this may sound, it weighs more.  The proportionality constant is 1/c², or 1.11 x 10^-17 kilograms per joule.  A fresh battery weighs more than a spent one, a wound-up alarm clock weighs more than a run-down one, etc.

Now that's way too small to measure for anything other than nuclear reactions, which is why it escaped everyone's notice for so long.  But it has been measured and experimentally verified for nuclear transformations all across the periodic table.

Now that's way too small to measure for anything other than nuclear reactions, which is why it escaped everyone's notice for so long.  But it has been measured and experimentally verified for nuclear transformations all across the periodic table.

#The universality of the speed of light.  (That is, special relativity.)

#The universality of the speed of light.  (That is, special relativity.)

Keep in mind that, under special relativity, it's not just space and time that need to be redefined.  The definitions of momentum and energy need to change also.  This is necessary so that the '''conservation of energy and momentum will be absolutely precise in all circumstances.'''

Keep in mind that, under special relativity, it's not just space and time that need to be redefined.  The definitions of momentum and energy need to change also.  This is necessary so that the '''conservation of energy and of momentum will be absolutely precise in all circumstances.'''

For example, under classical Newtonian mechanics, the kinetic energy of a moving mass is

For example, under classical Newtonian mechanics, the kinetic energy of a moving mass is

:<math>E = \frac{1}{2}mv^2\,</math>

:<math>E = \frac{1}{2}mv^2\,</math>

But under special relativity is is

But under special relativity is is

:<math>E = mc^2(\frac{1}{1 - v^2/c^2} - 1)\,</math>

:<math>E = mc^2\left(\frac{1}{1 - v^2/c^2} - 1\right)\,</math>

One can verify that, in the non-relativistic limit, the second of those equations converges to the first.

One can verify that, in the non-relativistic limit, the second of those equations converges to the first.

==History of Experimental Verification==

==History of Experimental Verification==

Because the change in mass arising from a given release of energy is so small (<math>1/c^2</math>, which 0.11x10^-16 kilograms per joule), it is essentially impossible to check this equation for normal processes.  For example, a flashlight battery loses about 1 picogram of mass when it discharges, and the resultant atoms from the detonation of 1 kilogram of TNT weigh 47 nanograms less than the TNT.  Even if all the particles of smoke and gas could be collected reliably, the difference couldn't be detected.

Because the change in mass arising from a given release of energy is so small (<math>1/c^2</math>, which 1.11 x 10^-17 kilograms per joule), it is essentially impossible to check this equation for normal processes.  For example, a flashlight battery loses about 1 picogram of mass when it discharges, and the resultant atoms from the detonation of 1 kilogram of TNT weigh 47 nanograms less than the TNT.  Even if all the particles of smoke and gas could be collected reliably, the difference couldn't be detected.

Measuring the effect requires process that release vastly more energy than ordinary chemical processes.  The discovery of Radium and Polonium around 1898 gave a tantalizing hint that there were processes that released far more energy than chemical processes could account for.  These elements continuously released measurable heat, and also glowed in the dark.

Measuring the effect requires process that release vastly more energy than ordinary chemical processes.  The discovery of Radium and Polonium around 1898 gave a tantalizing hint that there were processes that released far more energy than chemical processes could account for.  These elements continuously released measurable heat, and also glowed in the dark.