E = mc2 – Deriving units

Einstein is Relatively Special

Recently the question of the units of energy in Einsteins E = mc² came up in conversation. It seems the “meters squared” caused confusion. So here’s a short derivation of the units for energy.

The speed of light in a vacuum is denoted ‘c’ (lower case).  The mass of a particle is ‘m’. E is energy.
The units in E = mc² are derived as follows:
E = mc²    (Eqn. 1)
E = Energy, measured in units of Joule
m = mass, measured in units of kg
c = speed, measured in units of ms-1 (meters per second)
Note that ‘^’ denotes “raised to the power” e.g. c² = “c raised to the power 2” i.e. squared.

Taking the left hand side of the equality (the Energy):
Energy = force x distance    (Eqn. 2)
Force = mass x acceleration    (Eqn. 3)
Combining Eqn2 & Eqn.3 gives
Energy = mass x acceleration x distance    (Eqn. 4)
Substituting units into Eqn. 4 gives
Joule = kg x ms^-2 x m    (Eqn. 5)
Joule = kg m² s^-2    (Eqn. 6)

Taking the right hand side of the equality (mc²):
mc² = mcc    (Eqn. 7)
Substituting units into Eqn. 7 gives
= kg ms^-1 ms^-1    (Eqn. 8 )
= kg m² s^-2 (Eqn. 9)

Now lets bring both sides together:
Recall Eqn. 1: E = mc²
Substituting the left hand side (Eqn. 6) and the right hand side (Eqn. 9) gives:
kg m² s^-2 = kg m² s^-2
Q.E.D.

P.S. There is a relativistic form of E = mc² which deals with particles travelling close to the speed of light.