Importance of Newton's Second Law of Motion

The second law's real contribution is turning force into a precise, MEASURABLE, mathematical quantity: the rate of change of linear momentum, $\vec{F}=\frac{d\vec{p}}{dt}$.

It is essential to remember this momentum form as the fundamental statement, and NOT simply $\vec{F}=m\vec{a}$, because momentum $\vec{p}=m\vec{v}$ can change either because the velocity changes or because the mass itself changes (or both). Expanding the derivative using the product rule, $\vec{F}=\frac{d(m\vec{v})}{dt}=m\frac{d\vec{v}}{dt}+\vec{v}\frac{dm}{dt}=m\vec{a}+\vec{v}\frac{dm}{dt}$. Only when the mass is constant, i.e. $\frac{dm}{dt}=0$, does this collapse to the familiar $\vec{F}=m\vec{a}$. For a system like a rocket, where the body continuously loses mass (burnt fuel is ejected) while its velocity also changes, BOTH terms on the right must be kept -- $\vec{F}=m\vec{a}$ alone would be wrong.

This correction changes what counts as the 'fundamental' kinematic quantity of dynamics: it is MOMENTUM, not velocity, that a force directly changes. This matters because the effect of a force can show up as a change in a variable-mass system's mass distribution and not merely as a change in speed or direction. …