Why does the formula for kinetic energy have a half in it?

As you know energy is conserved so if the planet orbits the sun its potential energy is constantly converted to kinetic and vice versa. Our question is how to represent mechanical Energy through the notion of speed? Lets assume following example: We release a test body from the hight H and let it free fall to the groung. read more

The full, relativistic formula for kinetic energy of a body with rest mass [math]m_0[/math] travelling at velocity v is [math]m_0\,c^2\left(\gamma-1\right)[/math], where [math]\gamma[/math] is the Lorentz factor, [math]\frac{1}{\sqrt{1-v^2/c^2}}[/math] with c being the speed of light. read more