In Einstein's theory of relativity, we can derive that

E(v)=(mc^2)/(sqrt(1-(v^2/c^2)))

whereE(v)is the energy of an object with "Rest mass"mand velocityv.

Let us analyze this more closely.

First find the linear approximation to the functionf(u)=(mc^2)/(sqrt(1-u)) at u=0.

Using this approximation, and substitutingu=v^2/c^2, we can obtain an approximation for E(v)which is valid for small velocitiesv.

The approximation you obtain should have two terms. One of which is the famousE=mc^2(representing the resting energy) and the other should be the classical kinetic energy of the object.

The local linearization of(mc^2)/(sqrt(1-u)) at u=0 is ?

So we get that E(v)