= Solution
\b[The familiar result follows if the <force laws> have <Galilean invariance>; the three abstract <Newton's laws of motion> do not guarantee that invariance.] <Isaac Newton>'s Corollary V correctly identifies a central feature of ordinary <Newtonian mechanics>, but its general physical scope requires assumptions about interactions as well as about the relation between <force> and <acceleration>.<The Mathematical Principles of Natural Philosophy by Isaac Newton (1846 edition)>[Axioms, Corollary V and the accompanying equal-acceleration argument]{ref}
Take a constant <velocity> $\mathbf U$ and a <Galilean transformation>
$$
\mathbf x_i'=\mathbf x_i-\mathbf U t,\qquad t'=t.
$$
Then $\ddot{\mathbf x}'_i=\ddot{\mathbf x}_i$, $\mathbf x_i'-\mathbf x_j'=\mathbf x_i-\mathbf x_j$, and $\dot{\mathbf x}'_i-\dot{\mathbf x}'_j=\dot{\mathbf x}_i-\dot{\mathbf x}_j$. Any interaction depending only on these relative quantities, with unchanged parameters, supplies the same <forces> for the transformed initial data. <Newton's second law> consequently gives the same relative <accelerations>; uniqueness of the resulting <initial value problem> supplies the same relative histories. An inverse-square central interaction is a standard example. <Newton's third law> helps establish the common <centre of mass> result, but the boost argument principally needs invariant interactions and the second law.
Here is a counterexample to deduction from the three laws alone. Give two equal <masses> a preferred <inertial frame> and write $\mathbf r=\mathbf x_1-\mathbf x_2$, $\mathbf V=(\dot{\mathbf x}_1+\dot{\mathbf x}_2)/2$. For fixed $k>0$ and speed scale $u_*>0$, specify
$$
\mathbf F_{12}=-k\left(1+\frac{|\mathbf V|^2}{u_*^2}\right)\mathbf r,
\qquad \mathbf F_{21}=-\mathbf F_{12}.
$$
The <forces> are equal, opposite and central, the <centre of mass> is inertial, and each body obeys $m\ddot{\mathbf x}_i=\mathbf F_i$. An elementary body with no <force> is inertial. Nevertheless,
$$
\ddot{\mathbf r}=-\frac{2k}{m}\left(1+\frac{|\mathbf V|^2}{u_*^2}\right)\mathbf r.
$$
Boosting an isolated pair from $\mathbf V=0$ to $\mathbf V=\mathbf U$ changes its internal oscillation frequency. All three motion laws hold, but the proposed universal boost equivalence fails. The extra preferred structure in the <force law> is what makes the failure possible.
A passive <coordinate transformation> of this same model remains possible: transform the preferred frame's description too. That mathematical freedom does not establish an active <dynamical symmetry> which changes the physical pair's <velocity> relative to the preferred structure while holding the laws' background data fixed. Confusing those operations would hide the counterexample.
Finally, a change to rotating coordinates introduces <centrifugal forces> and <Coriolis forces>; their dependence on <position> or <velocity> can change internal experiments. Constant translational <velocity> has no analogous effect when the interactions have <Galilean invariance>. Corollary V expresses that physical symmetry, with the interaction assumption made explicit, rather than a consequence of action–reaction alone.
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