system
Given the following post, respond as if you're commenting on Reddit:
human
Why is it not E=1/2m c^2, like KE=1/2m v^2 ? Did Einstein leave off the 1/2?
gpt
Nah, Einstein didn't leave off the 1/2 and his equation actually includes it but in a subtle manner. Including object momentum, the full energy equation is,

* E^(2) = m^(2)c^(4)+p^(2)c^(2)

This is a fundamental result that arises from how the momentum vector is defined in special relativity. If the momentum is zero, you get back out good-old E=mc^(2). The energy is then the square root of this,

* E = √(m^(2)c^(4)+p^(2)c^(2))

If we divide out the mass, we're left with a function that looks like,

* f(x) = √(1+x^(2))

where x = p/mc. If you've done a bit of calculus you know you can Taylor series expand the function f(x) easily around the situation where x is small. Physically this means the momentum of the object is small compared to its mass. Perfect! This is the non-relativistic situation where we expect Newtonian mechanics and special relativity to match together. Then,

* f(x) = 1 + x^(2)/2 - x^(4)/8 + . . .

It's easy to see that for small x, the higher order terms are tiny and only get smaller. So let's throw them away and just keep the first two.

* f(x) ~ 1 + x^(2)/2

If we undo the substitution x=p/mc we get ultimately,

* E ~ mc^(2) + p^(2)/2m

The first term is just the energy you have because you have mass. The second term is the energy you have due to momentum. Or if we substitute the non-relativistic definition of momentum p=mv, we get the familiar,

* KE = mv^(2)/2

Therefore we've shown that the factor of 1/2 in the kinetic energy is well understood as a limit of kinetic energy in special relativity when the momentum is small and the object is not moving at a significant fraction of the speed of light.