The work of resultant forces equals the difference in kinetic energy

up:: Work

The work of a resultant force done upon a particle with mass $m$ equals the change of its kinetic energy.

$$W_L=\frac{m{v}_f^2}{2}-\frac{m{v}_i^2}{2}=\Delta K$$

Proof

Given a resultant force ${\vec{F}}_R$ acting on a particle of mass $m$, we have, from Newton’s Second Law that ${\vec{F}}_R=\frac{d\vec{p}}{dt}=m\frac{d\vec{v}}{dt}$.

From the definition of Work, we have that

$$W={\int }_L{\vec{F}}_R\cdot d\vec{l}={\int }_{t_0}^{t_1}{\vec{F}}_R\cdot \vec{v}dt=\frac{m}{2}{\int }_{t_0}^{t_1}\frac{d(\vec{v}\cdot \vec{v})}{dt}dt=\frac{m{v}_f^2}{2}-\frac{m{v}_i^2}{2}$$