Please derive e=mc^2 from Newton's equations?

I read "ON THE ELECTRODYNAMICS OF MOVING BODIES", By A. Einstein, June 30, 1905 the other day. In the section titled, 10. Dynamics of the Slowly Accelerated Electron, is the phrase:

Now if we call this force simply ``the force acting upon the electron,'' and maintain the equation--mass × acceleration = force ...

It became clear that E = mc² is really the energy (work) required to accelerate a mass to the speed of light in a vacuum.

The derivation is pretty simple. Based on Newton’s 2nd law of motion, Force is equal to mass times acceleration ( F = ma ). To calculate the energy, calculate the force of a mass accelerated from 0 to the speed of light:

Newton's second law of motion

F = ma = m * d/t/t >> Force (F) = mass * acceleration; Acceleration (a) = distance / time / time

E = F * d >> Energy (E) required to produce work = Force * distance ; 1 J = 1 N * m

F = m * d / t / t

Multiply both sides by d to get Energy on left side of equation.

F * d = m * d / t / t * d

E = m * d / t / t * d

E = m * d/t * d/t

E = m * (d/t) ^2

d/t = distance / time >> Velocity (v)

E = m * v^2

For velocity = speed of light in a vacuum, v = c

E = m * c^2

E = mc² >> energy to accelerate a mass to the speed of light.

Accelerating a mass to light speed must be like converting it to energy; in a vacuum.

Hope this helps.

1

You must mean it's similarity to Kinetic energy

E = 1/2 mv^2

In order to compensate for the apparent mass increase due to very high speeds we have to build it into our equations. We know that the mass increase can be accounted for by using the equation:

m = m'/sqrt(1 - v^2/c^2)

From this equation we know that mass (m) and the speed of light (c) are related in some way. What happens if we set the speed (v) to be very low? Einstein realised that if this is done we can account for the mass increase by using the term mc2 (the exact arguments and mathematics required to derive this are beyond the level of these pages, but it is a direct result of the equation above and speed of light being constant). Using this term we now have an equation that takes into account both the kinetic energy and the mass increase due to motion, at least for low speeds:

E = about mc^2 + 1/2 mv^2

This equation seems to solve the problem. We can now predict the energy of a moving body and take into account the mass increase. What's more, we can rearrange the equation to show that:

E - mc^2 = 1/2 mv^2 , for small v/c

This result is fine for low speeds, but what about speeds closer to the speed of light? We know that mass increases at high speeds, but according to the Newtonian part of the equation that isn't the case. Therefore, we need to replace the Newtonian part of the formula in order to make the equation correct at all speeds. How can we do this? We know that E - mc2 is approximately equal to the Newtonian kinetic energy when v is small, so we can use E - mc2 as the definition of relativistic kinetic energy:

Relativistic kinetic energy = E - mc^2

We have now removed the Newtonian part of the equation. Note that we haven’t given a formula for relativistic kinetic energy. The reason for this will become apparent in a moment. Rearranging the result shows that:

E = Relativistic kinetic energy + mc^2

It can now be seen that relativistic energy consists of two parts. The first part is kinetic and depends on the speed of the moving body. The second part is due to the mass increase and does not depend on the speed of the body. However, both parts must be a form of energy, but what form? We can simplify the equation by setting the speed (i.e. the relativistic kinetic energy) of the particle to be zero, thereby removing it from the equation:

E = Relativistic kinetic energy + mc^2

E = 0 + mc^2

so,

E = mc^2

Hi, Bob,

For the derivation of Einstein's "signature" equation wee Halliday, Resnick, and Walker (4th edition Chapter 42--Relativity) or any other college physics test, but it will not be based on Newton's view of the universe. Most texts will give you a nice comparison of the two views, but they do not "inter-relate in the way you are suggesting.

Your question is a bit like asking us to derive an orange from an apple.

-Fred

It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned above.

You do realize that relativity is different from Newtonian mechanics, right?

Not much gets by you does it? Apparently one or the other IS wrong. And it's Newton. Newtonian mechanics can be derived as a special case (low energies) or relativity. Relativity, however, can not be derived from classical mechanics.

Derivation Of E Mc2

E Mc2 Derivation

Bob,

Even Einstein didn't derive his eq from Newton's. He used those of Maxwell & Lorentz. Short answer:

Can't be done.

(Newtonian mechanics) - (Newtonian mechanics) + (Einstein)^(inertial reference frame) => E=mc^2.

"E" is a colloquial form of "he".

= means is; m is for "meh"

"meh" added to c makes "messy".

^ is an arrow upward pointing to God.

The 2, of course, is "too".

So E = mc^2 indicates God, saying "he is messy, too. [hence the discrepancy between what we see and what's really going on in the world.]"