The energy of matter is $ mc^2 $ where $ m $ is the mass of the matter and $ c $ is the velocity of light. We consider an example of a Newtonian system accelerating a mass $m $ from zero to velocity $ v $, where $ v \ll c $. The energy given up by the Newtonian system is $ mv^2 $. We also show that the work done by the accelerating system is half the energy given up by the accelerating system.

We know that the energy of matter is $ mc^2 $, i.e., its mass times the square of velocity. We say that a ball of mass $ m $ translating at velocity $ v $ has a kinetic energy of $ \frac{1}{2} {mv^2} $. But, isn’t its energy $ mv^2 $? We think so. How do we reconcile the fact that the work, i.e. energy expended, which is $ \int F dx $ required to bring the energy of the ball up to $ mv^2 $, when only half that amount of work is required?

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What is the Difference Between Energy and Kinetic Energy

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