We know of the equation that the kinetic energy of a body is:

$$K=\frac{m\_0c^2}{\sqrt{1-\frac{v^2}{c^2}}}-m\_0c^2 \ \ ,$$

where $m\_0c^2$ is the rest mass energy. If $v$ is very small, then:

$$\frac{v^2}{c^2} \to 0 \ \ ,$$

then we get:

$$K=m\_0c^2-m\_0c^2=0 \ \ .$$

Does that mean a moving body has *no* kinetic energy? How is it possible? We surely know the kinetic energy of that body should be $\frac{1}{2}mv^2$ according to newtonian mechanics.

Also, I think I have a major misconception regarding $E=mc^2$. I think of $E$ as the total energy of the body of mass $m$ (this is what our teacher taught us). But then does this mean the potential + kinetic energy of a body($E$) is always $mc^2$? Even if the body is stationary, how does it still have energy $mc^2$? If this is a great misconception, please enlighten me since our teachers taught us wrongly and they can't even answer my questions. All they do is use the formulae to solve problems.