\vec F = \frac{\hbox{d}(m\cdot \vec v)}{\hbox{d}t} = m \frac{\hbox{d} \vec v}{\hbox{d}t} + \vec v \underbrace{\frac{\hbox{d}m}{\hbox{d}t}}_{=0, m=\hbox{const}}  = m \frac{\hbox{d} \vec v}{\hbox{d}t} +   0 \equiv m\cdot \vec a