Reciprocal impulses do not determine a collinear impact: the same incoming data admit the sticking outcome $v^{+}=0$ and the elastic outcome $(v_1^{+},v_2^{+})=(-3,1)\mathrm{m}{\mathrm{s}}^{-1}$ with different impulses and energies

Common data: $M=8$, $\mu =\frac{3}{2}$, $V=\frac{2\cdot 3+6\cdot (-1)}{8}=0$, $d^{-}=-4n$, $\parallel d^{-}{\parallel }_2^2=16$, $K^{-}=12\mathrm{J}$.

For each candidate outgoing pair below, define $x_i(t)=t{v}_i^{-}$ for $t\le 0$ and $x_i(t)=t{v}_i^{+}$ for $t>0$. Both position curves are continuous on $I$, affine and hence $C^2$ on either side of $0$, and take values in the declared full domain $Q$. They have precisely the specified one-sided velocities, and $m_i{x}_i^{\mathrm{\prime }\mathrm{\prime }}=0=F_i(t,x,x^{\mathrm{\prime }})$ away from $0$. Thus all trajectory conditions of the isolated reciprocal impulse model hold for both candidates; only the reciprocal jumps remain to check. Moreover $⟨v_2^{-}-v_1^{-},n⟩=-4\mathrm{m}{\mathrm{s}}^{-1}<0$, so the common incoming data approach along $n$.

Outcome (a), sticking: $v_1^{+}=v_2^{+}=0$. Then $m_1(v_1^{+}-v_1^{-})=2(0-3)n=-6n$ and $m_2(v_2^{+}-v_2^{-})=6(0+1)n=+6n$; the impulse is $J=-6n\mathrm{N}\mathrm{s}$ and the total momentum is $0=P^{-}$.

Outcome (b), elastic: $v_1^{+}=-3n$, $v_2^{+}=+1n$. Then $m_1(v_1^{+}-v_1^{-})=2(-6)n=-12n$ and $m_2(v_2^{+}-v_2^{-})=6(+2)n=+12n$; the impulse is $J=-12n\mathrm{N}\mathrm{s}$ and the total momentum is $0=P^{-}$.

Together with the curves of step 1.2, steps 2.1 and 2.2 establish two complete runs of clause (i), with the same incoming data and different impulses ($-6n$ versus $-12n$) and outgoing velocities ($0$ versus $-3n$ and $+1n$). The first also satisfies restitution with $e=0$, and the second with $e=1$. Hence reciprocal impulse accounting alone does not fix the outgoing velocities, and the false statement fails.

Energy: by [L2] the changes are $\mathrm{\Delta }{K}_{(a)}=\frac{3}{2}\cdot \frac{1}{2}(0-16)=-12\mathrm{J}$ and $\mathrm{\Delta }{K}_{(b)}=\frac{3}{2}\cdot \frac{1}{2}(16-16)=0$; the outcomes are therefore also distinguished by their energy accounting, so no choice of impulse can be recovered from the momentum data and the two outcomes are genuinely different model histories.

Consequence: in the collinear isolated-impact model 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) (equivalently, the coefficient $e$, or a dynamical contact model producing the impulse) is an independent model datum; the reciprocal impulse equations constrain only the sum $m_1\mathrm{\Delta }{v}_1+m_2\mathrm{\Delta }{v}_2=0$. The same conclusion is recorded as clause 5 of 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.