In the formula e=mc2, how did Einstein know that from 1 part mass such an amount of energy of 299 792 458 x 299 792 458 m/s is released?

E=1/2 mv²

has to be adjusted. This is not noticeable for everyday speeds, but for speeds approaching the speed of light it is an important factor. The adjustment required according to the special theory of relativity is as follows:

This basically means that if an object is accelerated to a very high speed, part of the energy needed to reach that speed is not converted into speed, but into mass. That is, by the way, the reason that reaching the speed of light for an object with mass is impossible, since it needs an infinite amount of energy to do so and would reach an infinite mass.

The consequence is that we can convert mass into energy and vice versa. How do we derive the famous equation from that? The math behind that is a bit complicated, but we can understand it through a thought experiment like this (Einstein was a big fan of thought experiments):

Let’s consider a box floating stationary in space. A photon is emitted on one side of the box. That photon flies through the box from left to right. A photon is a particle that has no mass, but does have momentum. Because there is a law of conservation of momentum, when the photon arises on the left side of the box and moves to the right, the box will get an equal but opposite momentum to the left. When the photon is reabsorbed some time later on the right side of the box, the box will come to a stop again.

This poses a problem, however: there are no external forces acting on this system, so in principle the frame of reference of this system, its center of mass, must remain stationary. The photon has no mass, so it does not appear to contribute to the center of mass itself. Einstein solved this by assuming that the photon has an ‘equivalent mass’. We can derive this by forcing the center of mass of our box to remain constant.

For this we need the 2 impulses : that of the box and that of the photon. The following applies to the box:

p=Mv

and for the photon :

p=E/c

Those two must be equal, so

Mv=E/c

and the speed of the box is equal to the distance dx traveled, divided by the time dt :

Mdx/dt=E/c

The time it takes for the photon to bridge the length of the box, which we call L, is L/c, so :

M dx c/L=E/c

or

Mdx = EL/c²

Now if we assume that the ‘equivalent mass’ of the photon is m, then the center of mass is

Center of mass = (Mx₁+mx₂)/(M+m)

where x1 and x2 are the positions of the box and photon at the start of the experiment. We want this center of mass to remain constant, so the center of mass before and after the experiment must be equal, so :

(Mx₁+mx₂)/(M+m) = (M(x₁-dx)+m(x₂+L))/(M+m)

So after some simplification we get:

0 = mL-Mdx

or

Mdx = mL

We now plug this into the earlier equation of the impulses, which yields:

mL c/L = E/c

or

E=mc²

Thus, we have deduced that mass and energy are equivalent and can be converted into each other. The most striking practical example of a phenomenon where such a thing can be seen in reality is nuclear fission or fusion. That the conversion of a small amount of mass produces a gigantic amount of energy is most evident in stars, where mass is constantly being converted to energy by nuclear reactions that produce the enormous heat that makes stars burn.