Your explanation gives me a far better understanding of what you mean by “dimensional analysis”.
But before going into the “analysis” of the formula, I’ll explain the origin of the letter “U” in the formula.
Many years ago I tried to find out how gravity was related to mass, in other words “where” does gravity originate. I found that gravitational potential energy is usually given the symbol Ug. So I figured that if the U was good for gravity it should be good for me too. OK; back to the explanation of formula E=(U/k)*c.
Einstein’s formula indicates that mass and energy are “linked” together in such a way that you cannot have mass without the “potential energy” that exists within the mass. Nuclear activity can be used to change the “quality” of the mass in such a way that kinetic energy becomes very prominent.
The word “kinetic” comes from the Greek word kinētikos, meaning “of motion, which in turn traces to the verb kinein, meaning “to move.”
The atomic nucleus contains protons and neutrons, if there are too many neutrons in the nucleus then the nucleus becomes unstable, such condition means that the atom is “radioactive,” some radioactive atoms can be bombarded with more neutrons, when that happens the atomic nucleus becomes more radioactive and may break apart; when an atom breaks apart the broken parts fly off in all directions at a tremendous speed, such reaction is called fission and the speeding particles are known by the term “intrinsic kinetic energy.”
The formula E=(U/k)*c describes this phenomenon using mathematical symbols.
E= amount of kinetic energy. (J), which is the standard unit of measurement for kinetic energy. It is equivalent to 1 kg * m2/s2. In other words I am using the calculation of 1 kg * 1kg^2 = 1kg / 1s^2 = 1kg seconds of energy as a standard value for Joules of kinetic energy.
I am using the letter U as the potential (for) kinetic energy. The “potential” is also measured in Joules. To see how much “potential” kinetic energy there is in a one kg of mass you can do the following calculation. U = (299,792,458 * m^2)
c = constant, which is derived from the speed of light in a vacuum and is 299,792,458 meters per second.
k = the “intrinsic potential (for) kinetic energy” within the quantity of mass that is being scrutinized. The numeric quantity is the same amount as the m in the same calculation.
m = arbitrary quantity of mass that is being studied, and has the same numeric quantity as k.
Calculation potential energy for one kg mass is as follows.
The first part gives us the amount of “potential” for energy.
The second part gives us the intrinsic kinetic energy.
U = (299,792,458 * 1^2) and / 1 = 299,792,458
The total energy quantity then can be calculated with the following formula.
E = (U * m^2) /k) * c.
Energy for one kilogram of mass will become:
E = (299,792,458 * 1^2) and / 1 = 299,792,458) * 299,792,458 = 8.987551787368176*10^16 m2/s2.
For one kg of mass we can leave the m2/s2 out of the calculation because they do not contribute anything to the numeric outcome.
For greater mass numbers they may be necessary in order to clarify the logic of the computation.