Work and Kinetic Energy

Work done on an object by application of external force $\vec{F} (t)$ is given by:

W = \int_{t_{1}}^{t_{2}} \vec{F} (t) \cdot d \vec{r} (t)

W = \int_{t_{1}}^{t_{2}} \frac{d \vec{p}}{d t} \cdot \vec{v} (t) d t = \frac{1}{2} m v_{2}^{2} - \frac{1}{2} m v_{1}^{2} = K_{2} - K_{1}

Defining $K = \frac{1}{2} m v^{2}$ as kinetic energy, we see that the force acting on the system changes the kinetic energy of the system.