Answer:
[tex]W=K_f-K_i[/tex]
Explanation:
The work done on a particle by external forces is defined as:
[tex]W=\int\limits^{r_f}_{r_i} {F\cdot dr} \,[/tex]
According to Newton's second law [tex]F=ma[/tex]. Thus:
[tex]W=\int\limits^{r_f}_{r_i}{ma\cdot dr} \,\\[/tex]
Acceleration is defined as the derivative of the speed with respect to time:
[tex]W=m\int\limits^{r_f}_{r_i}{\frac{dv}{dt}\cdot dr} \,\\\\W=m\int\limits^{r_f}_{r_i}{dv \cdot \frac{dr}{dt}} \,[/tex]
Speed is defined as the derivative of the position with respect to time:
[tex]W=m\int\limits^{v_f}_{v_i} v \cdot dv \,[/tex]
Kinetic energy is defined as [tex]K=\frac{mv^2}{2}[/tex]:
[tex]W=m\frac{v_f^2}{2}-m\frac{v_i^2}{2}\\W=K_f-K_i[/tex]
