Consequences of the adopted impact and open-mass models: continuity of total momentum, the smooth-force exclusion, the restitution solution with $\mathrm{\Delta }K=-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$, and the open-body and rocket laws

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), inherited from the statement of Isolated reciprocal impulse with full-vector restitution: momentum jumps $\pm J$, conservation of total momentum, the restitution solution and the kinetic-energy change $-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$, whose distributional jump lemma uses it; the algebraic clauses used here do not otherwise use choice. Adopt the impact and open-mass extensions of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting, with $m_1,m_2>0$, an interval $I$ with nonempty interior, $t_{\ast }\in \mathrm{int}I$, an impulse motion $x=(x_1,x_2):I\to Q$ run in the isolated reciprocal impulse model at $t_{\ast }$ with impulse $J$ and one-sided velocities $v_i^{\pm }$, and write $M:=m_1+m_2,\mu :=\frac{m_1{m}_2}{M},P(t):=m_1{x}_1^{\mathrm{\prime }}(t)+m_2{x}_2^{\mathrm{\prime }}(t)(t\in I\setminus \{t_{\ast }\})\mathrm{.}$ Only the one-sided limits of $P$ at $t_{\ast }$ are used; no derivative of the position curves at the contact time is assumed. Then:

1. (Momentum continuity.) The total momentum is continuous across the impact, $P(t_{\ast }^{+})=P(t_{\ast }^{-}),\text{equivalently}{\sum }_{i=1}^2{m}_i(v_i^{+}-v_i^{-})=0,$ and the impulse delivered to label $1$ is removed from label $2$: $J=m_1(v_1^{+}-v_1^{-})=-m_2(v_2^{+}-v_2^{-})$.
2. (Smooth-force exclusion.) Suppose this same candidate motion $x$ from clause 1 also satisfies the smooth force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance and The associated first-order system, motions, maximal motions and the Newtonian flow of a force model on $I$, with the force maps continuous there. Then the momentum curves $p_i=m_i{x}_i^{\mathrm{\prime }}$ are of class $C^1$ on $I$; in particular $p_i(t_{\ast }^{+})=p_i(t_{\ast }^{-})(i=1,2)\mathrm{.}$ Clause 1 identifies the jump of $p_1$ with the same impulse $J$, so $J=0$: no candidate motion carrying a nonzero isolated impulse can also be a motion of that smooth-force class. The finite-jump model of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (i) is therefore a genuine extension of the smooth-force branch, not one of its cases.
3. (Collinear restitution solution.) Suppose in addition that the contact is collinear with contact direction $n$ and coefficient of restitution $e\in [0,1]$ (Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (ii)). Then $J=\mu (1+e)(v_2^{-}-v_1^{-}),$ $J$ is parallel to $n$, the centre-of-mass velocity $V:=M^{-1}(m_1{v}_1^{-}+m_2{v}_2^{-})$ is unchanged across the impact, and the post-impact velocities are $v_1^{+}=V+\frac{m_2}{M}e(v_2^{-}-v_1^{-}),v_2^{+}=V-\frac{m_1}{M}e(v_2^{-}-v_1^{-})\mathrm{.}$
4. (Kinetic energy of the impact.) The kinetic-energy change across the impact is $\mathrm{\Delta }K:=\frac{m_1\parallel v_1^{+}{\parallel }_2^2+m_2\parallel v_2^{+}{\parallel }_2^2}{2}-\frac{m_1\parallel v_1^{-}{\parallel }_2^2+m_2\parallel v_2^{-}{\parallel }_2^2}{2}=-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2\le 0,$ with equality if and only if $e=1$ or $v_2^{-}=v_1^{-}$. For $0\le e<1$ the contact model is dissipative: the total kinetic energy of the two labels is not conserved across the impact, and the deficit $\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$ is a quantity of the adopted contact law. This item identifies it with no measured heat, temperature, sound or deformation observable.
5. (Momentum balance alone does not determine the impact.) In the collinear case with $d^{-}:=v_2^{-}-v_1^{-}$ and $d_n:=⟨d^{-},n⟩<0$, the impulse equations of clause 1 alone leave the scalar impulse magnitude free: for every real $J_n$ with $J_n\le \mu d_n$ (so that the contact is separating or grazing) the velocities $v_1^{+}=v_1^{-}+\frac{J_n}{m_1}n,v_2^{+}=v_2^{-}-\frac{J_n}{m_2}n$ satisfy the reciprocal impulse equations with impulse $J=J_{n}n$ and preserve the total momentum, and their kinetic-energy change is $\mathrm{\Delta }K(J_n)=-J_n{d}_n+\frac{J_n^2}{2\mu },$ whose minimum over the admissible range is $\mathrm{\Delta }K(\mu d_n)=-\frac{\mu }{2}{d}_n^2$ at the perfectly inelastic value $J_n=\mu d_n$ (that is, $e=0$). The value of $J_n$ is fixed only by the contact law of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (ii), which selects $J_n=\mu (1+e)d_n$; in particular momentum balance alone does not determine the post-impact velocities.
6. (Open-body and rocket laws.) Under Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (iii), with differentiable $m:I\to (0,\mathrm{\infty })$ and $v:I\to {\mathbb{R}}^3$, $F:I\to {\mathbb{R}}^3$, $r_{\mathrm{i}\mathrm{n}},r_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to [0,\mathrm{\infty })$, and $w_{\mathrm{i}\mathrm{n}},w_{\mathrm{o}\mathrm{u}\mathrm{t}}:I\to {\mathbb{R}}^3$ satisfying the material balance identities, one has on $I$ $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{i}\mathrm{n}}(w_{\mathrm{i}\mathrm{n}}-v)-r_{\mathrm{o}\mathrm{u}\mathrm{t}}(w_{\mathrm{o}\mathrm{u}\mathrm{t}}-v)\mathrm{.}$ In particular: (a) if $r_{\mathrm{i}\mathrm{n}}=r_{\mathrm{o}\mathrm{u}\mathrm{t}}=0$ then $m{v}^{\mathrm{\prime }}=F$ and $(mv{)}^{\mathrm{\prime }}=F$; (b) if $r_{\mathrm{i}\mathrm{n}}=0$ and $r_{\mathrm{o}\mathrm{u}\mathrm{t}}=-m^{\mathrm{\prime }}>0$ (mass loss) then $m{v}^{\mathrm{\prime }}=F+r_{\mathrm{o}\mathrm{u}\mathrm{t}}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}})=F-m^{\mathrm{\prime }}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}}),$ so the thrust is $T:=r_{\mathrm{o}\mathrm{u}\mathrm{t}}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}})=-m^{\mathrm{\prime }}(v-w_{\mathrm{o}\mathrm{u}\mathrm{t}})$, and the closed-system law $(mv{)}^{\mathrm{\prime }}=F$ fails unless $w_{\mathrm{o}\mathrm{u}\mathrm{t}}=0$; (c) if $r_{\mathrm{o}\mathrm{u}\mathrm{t}}=0$ and $w_{\mathrm{i}\mathrm{n}}=0$ then $m{v}^{\mathrm{\prime }}=F-r_{\mathrm{i}\mathrm{n}}v$: capture contributes the acceleration $-(r_{\mathrm{i}\mathrm{n}}\mathrm{/}m)v$, opposing $v$, while $(mv{)}^{\mathrm{\prime }}=F$ still holds.

• Why the two models are kept separate from the smooth branch. Clause 2 excludes a nonzero momentum jump for a whole-interval $C^2$ solution of the regular force equations. The regular force map can remain continuous, even identically zero, in an impulse model; the finite jump is additional singular forcing. A bounded regular force integrated over intervals shrinking to $t_{\ast }$ contributes zero in the limit. The contact law is an additional adopted datum, just as the open-body accounting of Adopted impact and open-mass extensions of the force branch: the isolated reciprocal impulse model, the collinear restitution contact law and the open-body material accounting clause (iii) is.

• Comparison with the mathematically proved supplier. The endpoint algebra agrees with Isolated reciprocal impulse with full-vector restitution: momentum jumps $\pm J$, conservation of total momentum, the restitution solution and the kinetic-energy change $-\frac{1-e^2}{2}\mu \parallel v_2^{-}-v_1^{-}{\parallel }_2^2$ clauses b-c. That supplier additionally assumes continuous time forces and impulse pairings; steps 1.1-1.5 here use the adopted finite jumps directly and do not require those stronger hypotheses.

• Units and scale. $[J]=\mu \mathrm{\ell }{\tau }^{-1}$, $[\mathrm{\Delta }K]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$; both are model quantities. The coefficient $e$ is dimensionless model data; its empirical determination is a separate experimental task and is not asserted here.