Explanation:
The particle travels between two positions [tex]x_i[/tex] and [tex]x_f[/tex], this can be considered as the result of adding infinite elementary displacements. Therefore, the total work done by force in this displacement is given by:
[tex]W=\int\limits^{x_f}_{x_i} {F\cdot} \, dx[/tex]
According to Newton's second law:
[tex]F=ma=m\frac{dv}{dt}[/tex]
Recall that [tex]\frac{dx}{dt}=v[/tex] and [tex]K=\frac{mv^2}{2}[/tex]. So, we have:
[tex]W=\int\limits^{x_f}_{x_i} {ma\cdot} \, dx\\W=m\int\limits^{x_f}_{x_i} {\frac{dv}{dt}\cdot} \, dx\\W=m\int\limits^{v_f}_{v_i} {v\cdot} \, dv\\\\W=m[\frac{v_f^2}{2}-\frac{v_i^2}{2}]\\\\W=K_f-K_i\\W=\Delta K[/tex]
