Recently the question of the units of energy in Einsteins E = mc^{2} 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 = mc^{2} are derived as follows:

E = mc^{2} (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. c^{2} = “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 m^{2} s^{-2} (Eqn. 6)

Taking the right hand side of the equality (mc^{2}):

mc^{2} = mcc (Eqn. 7)

Substituting units into Eqn. 7 gives

= kg ms^{-1} ms^{-1} (Eqn. 8 )

= kg m^{2} s^{-2} (Eqn. 9)

Now lets bring both sides together:

Recall Eqn. 1: E = mc^{2}

Substituting the left hand side (Eqn. 6) and the right hand side (Eqn. 9) gives:

kg m^{2} s^{-2} = kg m^{2} s^{-2}

Q.E.D.

P.S. There is a relativistic form of E = mc^{2} which deals with particles travelling close to the speed of light.

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