work-energy theorem

The principle of work and kinetic energy states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle. This is essentially just a rephrased Newton's second law of motion, which says that the sum of all the forces acting on a particle determines the rate of change of momentum of the particle (and subsequently its motion).

Let's start by looking at the work done by all the forces acting (the net work) on a particle as it moves over an infinitesimal displacement. As per the definition of work, \[dW_{\text{net}}=\vec{F}_{\text{net}}\cdot d\vec{s}\]. Substituting Newton's second law, \[dW_{\text{net}}=m \left( \frac{d\vec{v}}{dt} \right)\cdot d\vec{s}=m d\vec{v}\cdot \left( \frac{d\vec{s}}{dt} \right)=m\vec{v}\cdot d\vec{v}\] (the last step substituted \[\frac{d\vec{s}}{dt}=\vec{v}\] and used the commutative property of the dot product). Then, we integrate between any two points on the particle's trajectory to get \[W_{\text{net},AB}=\int_{A}^{B}m\vec{v}\cdot d\vec{v}=\biggl[ \frac{1}{2}mv^{2} \biggr]_{A}^{B}=K_{B}-K_{A}=\Delta \text{KE}\]. Thus, we've shown that the net work done on a particle is equal to the change in its kinetic energy.