Variable-mass momentum balance: from $m^{\mathrm{\prime }}=r_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $(mv{)}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}{w}_{\mathrm{i}\mathrm{n}}-r_{\mathrm{o}\mathrm{u}\mathrm{t}}{w}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ to $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$

Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $m:I\to (0,\mathrm{\infty })$ and $v:I\to {\mathbb{R}}^3$ be differentiable (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral), and let $F:I\to {\mathbb{R}}^3$, $r_{\mathrm{i}\mathrm{n}},r_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to [0,\mathrm{\infty })$ and $w_{\mathrm{i}\mathrm{n}},w_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to {\mathbb{R}}^3$ be maps. Suppose the two material balance identities

$m^{\mathrm{\prime }}(t)=r_{\mathrm{i}\mathrm{n}}(t)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(t),(mv{)}^{\mathrm{\prime }}(t)=F(t)+r_{\mathrm{i}\mathrm{n}}(t)w_{\mathrm{i}\mathrm{n}}(t)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(t)w_{\mathrm{o}\mathrm{u}\mathrm{t}}(t)$

hold for every $t\in I$, where $mv$ denotes the pointwise scalar multiple. Then

$m(t)v^{\mathrm{\prime }}(t)=F(t)+r_{\mathrm{i}\mathrm{n}}(t)(w_{\mathrm{i}\mathrm{n}}(t)-v(t))-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(t)(w_{\mathrm{o}\mathrm{u}\mathrm{t}}(t)-v(t))(t\in I)\mathrm{.}$

In particular, if $F\equiv 0$, $r_{\mathrm{i}\mathrm{n}}\equiv 0$ and $r_{\mathrm{o}\mathrm{u}\mathrm{t}}\equiv r$ for a constant $r\ge 0$, then $m{v}^{\mathrm{\prime }}=-r(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)$: the acceleration is governed by the momentum carried away per unit time relative to the body, not by $(mv{)}^{\mathrm{\prime }}=0$.

The two balance identities are the adopted content of the material momentum accounting — the mathematical rendering of "gains $r_{\mathrm{i}\mathrm{n}}dt$ of matter with velocity $w_{\mathrm{i}\mathrm{n}}$, loses $r_{\mathrm{o}\mathrm{u}\mathrm{t}}dt$ with velocity $w_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and feels the external force $F$". They are not derived here from any closed-system law; the deduction below is their product-rule consequence.