system
Given the following post, respond as if you're commenting on Reddit:
human
Why does E=mc^2 not have the 1/2 coefficient that KE=1/2 mv^2 has?

I understand the 1/2 for many energy formulas comes from calculus integration. Why do some formulas like E=mc^2 not have the half, and what is the non-math way to explain the significance of having that coefficient (in layman's terms)
gpt
The full relativistic expression for the energy (combining rest energy and kinetic energy) for a particle of mass m and speed v is:

mc^(2) / sqrt{1-v^(2)/c^(2)}

When v is small compared to c, as it is for every material object we encounter in day-to-day life, that above expression takes the 
approximate form

mc^(2) +(1/2)mv^(2)

That's how the factor of 1/2 arises.  The first term is the rest energy, and the second term is classical kinetic energy.

If v gets larger, the next, slightly better approximation is

mc^(2) +(1/2)mv^(2)+(3/8)mv^(4)/c^(2)

so that next term has a 3/8 in it!