
Can you explain E= MC^{2} ?
  Answer 1:
This is Einstein's brilliant realization that
energy (E) and mass (M) are actually equivalent
stuff, but behave differently at different
temperatures. It was revolutionary at the time,
because he was the first to realize it! Now we
take it for granted, but we need it to understand
a variety of high energy processes in the
Universe! Let's say we have "stuff". At very
high temperatures (like millions and millions of
degrees, hotter than you can imagine) all the
"stuff" is just pure radiation  pure bright
light, and it is very hot. But if you somehow cool
this "stuff" then matter will start to condense
out of the radiation, like drops of water condense
out of cooling steam. So that is a way you can
think about what this equation is telling you.
Here's an example of how we use this equation
in physics: The helium atom actually weighs
just a tad bit less than the sum of the weights of
2 protons and 2 neutrons. This is because when
hydrogen is converted into helium by nuclear
fusion (like what happens inside the sun to make
it shine) a little bit of mass is actually
converted into pure energy. This amount of mass is
exactly equal to the amount of energy that is
released as heat and light that we measure! It's
just a little bit, but there is so much matter in
the sun, (2 x 10^30 kg of matter!) that it adds up
to a lot of heat and light. (Lucky for us on
Earth!) Here is another example: If you take
one particle of matter, say a proton, and combine
it with its antiparticle, the antiproton, what
you get is pure radiation, and the amount of
radiation is exactly equivalent to what you would
calculate from that equation E = Mc^2. By the same
reasoning, if you have pure radiation that is hot
enough, then equal amounts of matter and
antimatter can be spontaneously (all of a sudden,
all by themselves) created out of the radiation!
In the very early universe  like within the first
second after the birth of the universe  the
temperature was so hot that matter and antimatter
particles were being spontaneously created out of
the radiation "field". But as soon as they were
created, they would annihilate each other and turn
back into pure energy. We can understand this from
Einstein's equation. Fortunately for us,
however, there was just a little more matter than
antimatter, and so when the universe cooled enough
so that this creation process had to stop, there
was just matter and radiation left, and the
radiation cooled down as the universe expanded,
until today, when it is around 3 degrees above
absolute zero (about 270 Celsius, or 454
Fahrenheit).
  Answer 2:
E = M C^2 relates energy to mass. Specifically it
says that if you convert 1 g of mass to energy,
you get 9x10^20 ergs of energy  (To use this
just keep consistent units i.e. Joules, Meters,
Seconds, Kilograms, and Ergs, centimeters, grams,
seconds. It does not tell you how to do it 
however, you can measure the mass of a deuterium
atom and of a helium atom  and 2 deuterons weigh
more than 1 helium atom. So in fusion, if you
overcome the repulsion of the protons in the
deuterium and fuse them to make helium, the mass
difference is made up in energy  this slight
difference powers most of the stars... The
revolution was that energy and mass were formerly
considered to be fundamentally different
commodities. In a sense, this relation forged a
new understanding that mass and energy were in
many senses the same.
  Answer 3:
Let me start by saying that I encourage you to
keep up your questions and look into the answers!
There are a lot of great books on the special
theory of relativity (which includes that
equation), so you might try going to a bookstore
or library. First, let's identify the variables
in the equation. M is the mass of the object that
we are considering. c is the speed of light moving
through a vacuum (such as outer space). E is the
energy of the object. What is revolutionary about
this equation is that it means the mass of an
object can be turned into energy. Before Albert
Einstein discovered relativity, no one had ever
believed that mass and energy could be the same
thing. Before, physicists believed that mass
always stayed the same and that energy always
stayed the same; now we know that they can be
changed into each other. The nuclear reactions in
the sun happen precisely because they turn mass
into energy. Those are just the basics. I need
to tell you also that E=Mc^2 is often
oversimplified. To be more precise, we should
write E=Mc^2/(1v^2/c^2)^(1/2). Now we mean that M
is the mass of the object when it is at rest, and
v is the speed of the object. This is the correct
relativistic formula for energy. It includes both
the "mass energy" from the rest mass and the
"kinetic energy" (energy of motion: you might have
learned about it, since you are in 8th grade). The
other thing is that what we mean by "rest mass" is
often really energy in disguise. For example, if
we look at the mass of a nucleus, it is less than
the sum of the masses of the parts. That's because
the "potential energy" that holds the parts
together is negative, so E=Mc^2 tells us that the
mass should decrease by that amount. This is just
the start of the story, too, as it turns out that
most of the mass of the parts of a nucleus comes
from the kinetic and potential energies of their
parts!This is such a long and interesting story
that I urge you to get out and look into it yourself!
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