$$\int_i^f m\dfrac{d\vec{v}_{cm}}{dt}\cdot d\vec{r}_{cm} = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$

$$\int_i^f m\,d\vec{v}_{cm}\cdot\dfrac{d\vec{r}_{cm}}{dt} = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$

$$\int_i^f m\,\vec{v}_{cm}\cdot d\vec{v}_{cm} = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$

$$\int_i^f m\,{v}_{cm} dv_{cm} = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$

$$\dfrac{1}{2}m\,{v}_{cm,f}^2- \dfrac{1}{2}m\,{v}_{cm,i}^2 = \int_i^f \vec{F}_{ext} \cdot d\vec{r}_{cm}$$

$$\Delta K_{trans}= W_{cm}$$