e = mc2 = 1/2mv2 = ke

You bet it is! The condition is that v is much less than c.

The rest energy of a thing is E = mc2.

i recall seeing this proof when reading asimov.

i found this was a different source.

The total energy of a thing is gamma├Śmc2.

The kinetic energy is the difference of these, KE = (gamma – 1)mc2.

Recall that gamma = (1 – (v/c)2)-1/2. Expanding this in Maclaurin series it’s:

gamma = 1 + 1/2 (v/c)2 – 3/8 (v/c)4 + …

So KE = (1/2 (v/c)2 – 3/8 (v/c)4 + … )mc2

If v is much less than c, we can approximate this with just the first term, so

KE is approximately 1/2 (v/c)2 mc2 = 1/2 mv2

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