Re: How do I calculate the energy-mass relationship?

In 1905, Albert Einstein published a paper entitled “Does the Inertia of a 
Body Depend Upon Its Energy Content”, which included a derivation of the 
formula ‘energy is equal to mass times the speed of light squared’.  That 
derivation does not depend upon calculus.  I Found the article in 
Einstein: The Principle of Relativity; H. A. Lorentz, H. Weyl, and H. 
Minkowski; Dover Publications, Inc.; a 1952 unabridged and unaltered 
reprint of a 1923 translation of the original.

Essential Elements of Einstein’s Mathematical Basis for  E = mc2

I.  Let there be two inertial reference frames as follows:

    A.  Frame 1: with coordinates (x,y,z)

    B.  Frame 2: with coordinates (x’, y’, z’); where the x’, y’, and z’ 
axes are parallel to the x, y, and z axes of Frame 1 respectively.  Also, 
Frame 2 has a velocity with a magnitude of  v  and directed in the 
positive-x direction.

II.  Let there a be a body of mass m at rest in Frame 1 and let its energy 
relative to Frame 1 be E0.  Let the energy of the same body relative to 
Frame 2 be H0.

III.  Let this same body emit light of energy 1/2 L measured relative to 
Frame 1, in a direction making an angle phi with the x-axis.  
Simultaneously, let light of equal energy (1/2 L) be emitted in the 
opposite direction.  After light emission, let the energy of the body be 
E1 in Frame 1 and H1 in Frame 2.

IV.  To simplify the remainder of the derivation, I set phi equal to zero, 
in effect having the light emitted in the positive-x and negative-x 
directions.

V.  By the principle of conservation of energy, in Frame 1,

E0 = E1 + 1/2 L + 1/2 L              [equation 1]

or the energy of the body at rest in Frame 1 before light emission is 
equal to the energy of the body after emission plus the energy of the 
light emitted.

VI.  By the principle of relativity, energy is also conserved in Frame 2, 
or

H0 = H1 + 1/2 L [ (1 - v/c) / (1 - v2/c2)**(1/2) ]  + 1/2 L [ (1 + v/c) / 
(1 - v2/c2)**(1/2) ]          [equation 2]

Now, this is the first part of the derivation that may not be standard 
fare in high school physics.  The equation is further complicated because 
I did not use a fancy equation writer, thus the following shorthand is 
used in the equation:

  v2 - stands for velocity squared
  c2 - stands for the speed of light squared
  (1 - v2/c2)**(1/2) - is used to indicated the square root of  (1 - v2/c2)

If you need additional information to understand where equation 2 comes 
from, you may need to study Lorentz transformation as described in 
introductory textbooks on special relativity, such as the following:

Ray Skinner, Relativity for Scientists and Engineers, Dover Publications, 
1982

Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, W. H. 
Freeman and Company, 1966

Some high school and most college physics texts also cover special 
relativity to the level of detail needed to understand equation 2.

VII.  Einstein then equated the two differences of the form  H - E as 
follows

    H0 - E0  =  K0 + C                  [equation 3]

    H1 - E1  =  K1 + C                  [equation 4]

where C is an additive constant not changed by the emission of light from 
the body described above, and K0 and K1 represent kinetic energy of the 
body in Frame 1 and Frame 2, respectively.

In more detail, first combine equations 1 and 2,

   ( H0 - E0 ) =  ( H1 - E1 )  +   1/2 L{  [(1-v/c)/(1-v2/c2)**(1/2)] - 1 
} + 1/2 L {  [(1+v/c)/(1-v2/c2)**(1/2)] - 1 }

simplify to,

    (K0 + C) = (K1 + C)  +  L [ 1/(1-v2/c2)**(1/2)]

and then,

    K0 - K1  =  L [ 1/(1-v2/c2)**(1/2)]                   [equation 5]

VIII.  Finally, by a binomial series expansion,

    [ 1/(1-v2/c2)**(1/2)]  becomes 1/2 v2/c2, or equation 5 becomes

change in kinetic energy of the body is equal to 1/2  [L/c2]  v2, or

since kinetic energy is generally known as 1/2 m v2, it follows, given 
that m0 is the mass of the body before light emission and m1 is the mass 
of the body after light emission, that equation 5, now in the form

    K0 - K1  =  1/2 [L/c2] v2  =  1/2 [m0 - m1] v2          [equation 6], 
or

the change in mass of the body, from before to after light emission, must 
be equal to  L/c2, or

    L  =  [m0 - m1] c2                [ equation 7],

essentially identical to

Energy is equal to mass times the speed of light squared.

My apologies for such a lengthy reply.  I hope this helps.  I thought 
mention of intertial reference frames might also cause difficulty, but a 
careful explanation of inertial refererence frames could easily have 
doubled the length of this response.  The same references, though, have in 
depth discussions of inertial reference frames, without the use of 
calculus.