Deriving the equation -- Advanced

This page takes a formal approach to deriving Einstein's E = mc
2
using calculus and assumes a background knowledge
of basic physics, as well as relativity as related on the other pages in this series. The equation is also derived without
calculus on this page, and the terms used in the equation are explained here.

We start by noting that energy is the integral of force with respect to distance, so kinetic energy K can be defined by:

Where F is the force in the direction of displacement ds and s is the distance over which the force acts.

Using Newton's second law of motion force F can be shown as:

Thus the equation for kinetic energy K can now be shown as:

Note that the velocity limit is c (the speed of light). At c time dilation becomes 100% and distances in the direction of
motion shrink to zero, hence a body at this speed will not experience time or distance and so its velocity is set as the
upper limit.

We now integrate by parts :

To yield:

The result shows that the kinetic energy of a body is equal to the increase in its mass as a consequence of its relative
motion multiplied by c
2
. This can be rearranged to show:

If the kinetic energy is decreased so that K = 0 the body will be stationary, but will still possess energy m
0
c
2
. In other
words the body contains energy E
0
when stationary relative to its frame and will have mass m
0
. This is called the rest
mass. This is shown as:

where:

This, then, completes the derivation of E = mc
2
for a body at rest. For a moving body its total energy is given by:

Deriving the equation -- Advanced

This page takes a formal approach to deriving Einstein's E = mc
2
using calculus and assumes a background knowledge of basic
physics, as well as relativity as related on the other pages in this
series. The equation is also derived without calculus on this
page, and the terms used in the equation are explained here.

We start by noting that energy is the integral of force with
respect to distance, so kinetic energy K can be defined by:

Where F is the force in the direction of displacement ds and s is
the distance over which the force acts.

Using Newton's second law of motion force F can be shown as:

Thus the equation for kinetic energy K can now be shown as:

Note that the velocity limit is c (the speed of light). At c time
dilation becomes 100% and distances in the direction of motion
shrink to zero, hence a body at this speed will not experience
time or distance and so its velocity is set as the upper limit.

We now integrate by parts :

To yield:

The result shows that the kinetic energy of a body is equal to the
increase in its mass as a consequence of its relative motion
multiplied by c
2
. This can be rearranged to show:

If the kinetic energy is decreased so that K = 0 the body will be
stationary, but will still possess energy m
0
c
2
. In other words the
body contains energy E
0
when stationary relative to its frame and
will have mass m
0
. This is called the rest mass. This is shown as:

where:

This, then, completes the derivation of E = mc
2
for a body at rest.
For a moving body its total energy is given by: