Why Is Mass Halved in the Work-Energy Theorem Derivation?

[Scalise]

Hello,

Someone could explain me why in the derivation below the mass m is divided by 2 in the last step?:

$\int\vec{F}\cdot d\vec{s}=m\int\frac{d\vec{v}}{dt}\cdot\vec{v}dt=\frac{m}{2}\int \frac{d}{dt}(v^{2})dt$

 

[jbriggs444]

Hello,

Someone could explain me why in the derivation below the mass m is divided by 2 in the last step?:

$\int\vec{F}\cdot d\vec{s}=m\int\frac{d\vec{v}}{dt}\cdot\vec{v}dt=\frac{m}{2}\int \frac{d}{dt}(v^{2})dt$

The derivative $\frac{d}{dt}(v^{2})$ is equal to $2\vec{v}\cdot\frac{d\vec{v}}{dt}$