Why is kinetic energy defined as $(1/2)m v^2$?

Question

What is special about $(1/2)m v^2$ that makes physicists believe that it is a representation of kinetic energy?

Answer 1

It's the work done to accelerate a particle from rest to a final speed $v$, i.e. the energy "put in" to initiate motion.

Work $W \equiv \int_0 ^{v} \,F \, dr$, with $F = ma = m\frac{dv}{dt} = m \frac{dv}{dr}\frac{dr}{dt} = mv\frac{dv}{dr}$, so:

$$W = \int_0 ^{v} \,F \, dr=\int_0 ^{v} \, mv'\frac{dv'}{dr} dr = \int_0 ^{v} \, mv'\,dv' = \frac{1}{2}mv^2$$