Why does E=mc squared?

Answer 1

First start with momentum:

p = mv (or mass times velocity)

This is the classical version. Using the relativistic form of mass, replace m with γm0 where m0 is the rest mass and

γ = 1/√(1-v²/c²)

Force is the change in momentum with respect to time.

F = dp/dt = d(mv)/dt = mdv/dt + vdm/dt

Classically, dv/dt = a and dm/dt = 0 so F=ma.

Now let's calculate dm/dv on our relativistic momentum.

dm/dv = m0dγ/dv
dm/dv = m0d[1/√(1-v²/c²)]/dv
dm/dv = (m0v/c²)(1-v²/c²)^-1.5
dm/dv = (m0v/c²)γ³
m = m0γ = dm/dv c²/(vγ²)
m = dm/dv c²(1-v²/c²)/v
m = dm/dv (c²-v²)/v

F = mdv/dt + vdm/dt
F = [dm/dv (c²-v²)/v]dv/dt + vdm/dt
F = (c²-v²)/v dm/dv dv/dt + vdm/dt

By the chain rule, dm/dv dv/dt = dm/dt

F = (c²-v²)/v dm/dt + vdm/dt
Fdt = (c²-v²)/v dm + vdm

Now multiply by v=dx/dt (velocity is the change in position with respect to time).

Fdt dx/dt = (c²-v²) dm + v²dm
Fdx = (c²-v²)dm + v²dm
Fdx = c²dm

Fdx is just the change in kinetic energy. Since work is force times distance, a constant force applied over a short interval dx increases the kinetic energy by dE. So we have

dE = c²dm

Now you just integrate both sides.

E = mc²

Answer 2

if a body emits a certain amount of energy, then the mass of that body must decrease by a proportionate amount.

Equation E=mc² . Here E represents energy, m represents mass, and c² is a very large number, the square of the speed of light.

In 2005, the centennial of Einstein’s great year, a team made the most accurate test yet of his equation. They measured the tiny change in mass of radioactive atoms before and after the atoms emitted gamma-rays. And they measured the energy of the rays. The missing mass times c² equalled the energy of the rays to within 4 hundred-thousandths of one percent.

Answer 3

In physics, mass–energy equivalence is the concept that all mass has an energy equivalence, and all energy has a mass equivalence. Special relativity expresses this relationship using the mass–energy equivalence formula

E = mc2
where

E = the energy equivalent to the mass (in joules),
m = mass (in kilograms), and
c = the speed of light in a vacuum (celeritas) (in meters per second).
Several definitions of mass in special relativity may be validly used with this formula, but if the energy in the formula is rest energy then the mass must be the rest mass (also called the "invariant mass").

Origination of the formula is popularly attributed to Albert Einstein in 1905 in what are known as his Annus Mirabilis ("Wonderful Year") Papers, though Einstein was not the first to propose a mass–energy relationship, and the formula appeared in works predating Einstein's theory (see Contributions of others, below).

In the formula, c² is the conversion factor required to convert from units of mass to units of energy, i.e., the energy density. In unit-specific terms, E (joules or kg·m²/s²) = m (kilograms) multiplied by (299,792,458 m/s)2.

Answer 4

In physics, mass–energy equivalence is the concept that all mass has an energy equivalence, and all energy has a mass equivalence. Special relativity expresses this relationship using the mass–energy equivalence formula

E = mc^2

where

* E = the energy equivalent to the mass (in joules),
* m = mass (in kilograms), and
* c = the speed of light in a vacuum (celeritas) (in meters per second).

Several definitions of mass in special relativity may be validly used with this formula, but if the energy in the formula is rest energy then the mass must be the rest mass (also called the "invariant mass").

Origination of the formula is popularly attributed to Albert Einstein in 1905 in what are known as his Annus Mirabilis ("Wonderful Year") Papers, though Einstein was not the first to propose a mass–energy relationship, and the formula appeared in works predating Einstein's theory.

In the formula, c² is the conversion factor required to convert from units of mass to units of energy, i.e., the energy density. In unit-specific terms, E (joules or kg·m²/s²) = m (kilograms) multiplied by (299,792,458 m/s)2.

Conservation of mass and energy

The concept of mass–energy equivalence unites the concepts of conservation of mass and conservation of energy, allowing mass to be converted to forms of active energy (such as kinetic energy, heat, or light) while still retaining mass. Conversely, active energy in the form of kinetic energy or radiation can be converted to particles which have rest mass. The total amount of mass and energy in a closed system (as seen by a single observer) remains constant. Energy cannot be created or destroyed, and in all of its forms, trapped energy exhibits mass. In relativity theory, mass and energy are two forms of the same thing, and neither one appears without the other.

Answer 5

Einstein's famous equation, usually written as
E = m*c^2
where E is energy, m is mass, and c is the speed of light (300,000,000 meters per second),
describes how mass is really just another form of energy, and shows you how to convert between them.

In the explosion of the bomb dropped on Hiroshima, Japan, about 5 grams of mass (about the mass of a coin) disappeared off the face of the Earth, converted entirely to energy (in the form of photons and heat), wiping out an entire city and thousands of people in the process. There's a LOT of energy stored in mass.

Answer 6

Einstein's theory of relativity states that energy (e) equals mass (m) TIMES the speed of light (c) to the second power (squared).

Answer 7

E = the total amount of Energy
At the constant speed of light (c) time does not pass
(i.e. a photon which left a star one million light years from here took one million years to get here for us, but no time passed for the photon)
m = Mass

So if no time is passing, nothing is changing and therefore Mass and Energy must remain costant (this is the thought game Einstein used to rationalize this equation)

Answer 8

This is a physics question. The answer is not in the domain of mathematics but lies in the relationship between energy and mass. Your question belongs to the Physics forum.

Answer 9

I was going to say there is no "why", it's just a truth about the way energy and mass work in our universe. But I like Poetland's recounting of the thought-experiment. I had never heard that one, and it does help the intuition.

Answer 10

Because it was found that energy and mass are interchangeable (as in the atomic bomb) and this formula describes the observed relationship.