[download as pdf: EMC2_rev01.pdf]

In this note we present the derivation of Einstein formula using the framework of classical dynamics with the addition of the expression for the relativistic momentum .

A quick comparison is also made with the usual derivation of Newtonian kinetic energy.

We’ll consider a body with rest mass that starts from rest and reach the speed . It’s energy is equal to the work done to make the body reach its final speed .

since

In **Newtonian mechanics**, it is and, assuming the body’s mass doesn’t change, it is

That’s the classical **kinetic energy** (KE) acquired by the body.

In **relativity** we must use a different expression for the momentum

where

It can be noted that, in order to reconcile this new definition of momentum with the classic one (), it’s possible (even if a little controversial) to define the *relativistic mass* as the mass of a moving body so that we can write

In this case the integral is

We’ll calculate the integral in two different ways: **a)** with a substitution and using a trigonometric function for the ratio and **b)** without the substitution, keeping the original variables and following the usual rules for differentiation and integration.

# a) The “trig” route

The “*trig*” route presented in the following lines has some interesting geometric interpretations and makes the calculations easier. It’s based on a simple change of variable:

then and

meaning that

With these new expressions the integral becomes:

And the integration is almost straightforward:

Coming back to the original variables we have

So the relativistic kinetic energy is the excess of with respect to the invariant *rest energy* and it is

Above equation leads to Einstein well-known formula when we define (total energy) as the sum of the *kinetic energy* (the energy due to the speed acquired) and the intrinsic *rest energy* (the base energy due to the rest mass only, ).

The following figure shows the interesting geometric relations between the main quantities involved in the derivation:

As a bonus we get the important **energy-momentum** relation

that follows from the simple application of Pythagoras’ theorem to the triangle .

# b) Integration “as it is” and without substitution

We can also calculate the integral

without recurring to the previous substitution

Let’s first calculate the term :

So it is

We have reached the same result of the first calculation (with the “*trig*” substitution) but with some more effort and no evident geometric visualization and guidance.

*Last updated: 1 june 2018*