Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S27 / 1

Newton's first law says that a body retains uniform rectilinear motion, including rest as the zero-velocity case, unless impressed forces change that motion. Its physical content includes the existence of a common class of inertial frames: free bodies do not each require an unrelated frame chosen just to straighten their own trajectory. Uniformity refers to the inertial time parameter, and straightness to the spatial structure of those frames. Everyday slowing under friction is compatible with this claim because it involves an interaction, not an exception to inertia.

Newton's second law relates the impressed resultant force to change of momentum. In modern continuous-force notation,so for constant mass it becomes $m\mathbf{a}={\mathbf{F}}_{\mathrm{net}}$. An impulsive version is $\Delta \mathbf{p}=\int \mathbf{F}dt$. The change is directed along the impressed resultant, not necessarily along the body's current velocity. The original wording concerns change of quantity of motion; the differential equation is its modern continuous-time formulation.^[ref]

These are substantive physical claims only when mass, interactions and the inertial structure are constrained independently. If “force” merely abbreviates whatever equals $m\mathbf{a}$ in any arbitrarily chosen coordinates, the second law by itself predicts nothing. Actual applications combine the law with independently testable interaction laws, vector addition of forces, initial conditions and a specification of the inertial frames.

Showing that acceleration is unchanged by a uniform boost is necessary, but it is not enough to establish the Galilean covariance of Newton's second law. One must specify how time, positions, masses, forces and any background physical systems transform, and show that the actual force laws retain their form.

For a constant spatial rotation $R$, boost velocity $\mathbf{U}$ and translations, writeDifferentiation gives ${\mathbf{v}}^{\prime }=R\mathbf{v}-\mathbf{U}$ and ${\mathbf{a}}^{\prime }=R\mathbf{a}$. Assuming invariant inertial mass,Consequently the equation has the same form if the resultant physical force satisfies ${\mathbf{F}}^{\prime }=R\mathbf{F}$. For a pure boost, this reduces to ${\mathbf{F}}^{\prime }=\mathbf{F}$. More explicitly, if a force is represented by a function of positions, velocities and other physical data, those arguments and data must transform so that its primed value is $R$ times the old value. It is this condition on the complete dynamical model, not the typography of $\mathbf{F}=m\mathbf{a}$, that establishes covariance.

An instantaneous inverse-square interaction depending on the separation of two particles passes this test: their common boost cancels from the separation, and a rotation rotates the force vector. A resistance law $\mathbf{F}=-\zeta (\mathbf{v}-\mathbf{u})$ also passes when the medium's velocity $\mathbf{u}$ transforms with the body. In contrast, keeping $\mathbf{F}=-\zeta \mathbf{v}$ in every boosted frame while silently resetting the same medium to rest changes the physical problem. That model has not been shown covariant by noting ${\mathbf{a}}^{\prime }=\mathbf{a}$.

Likewise, the full Maxwell equations are not made Galilean covariant merely because some equation of particle motion is written as force equals mass times acceleration. The covariance of a framework has to be checked together with its interaction laws. Newton's own Corollary V expresses the corresponding equivalence of uniformly translating, nonrotating systems for the mechanics under consideration.^[ref]

Within an already specified inertial frame, with constant mass, substituting zero resultant force into Newton's second law does give zero acceleration. In that limited mathematical sense, force-free uniform motion is a special case. The reason for retaining Newton's first law is that its physical role is broader than this substitution.

The second law does not, just by its algebra, identify the coordinate systems in which acceleration measures the response to physical interactions. A free body described in coordinates $x^{\prime }=x-\frac{1}{2}A{t}^2$ has ${\ddot{x}}^{\prime }=-A$, despite experiencing no impressed interaction. If the second law is restricted to inertial frames, the first law supplies a criterion for that restriction and asserts that a common inertial structure exists. If instead one introduces a fictitious force $-mA$ to preserve the equation in the accelerated coordinates, that coordinate-dependent bookkeeping should not be confused with a newly discovered physical interaction.

Nor can the first law simply be used as a circular definition in which “free” means “unaccelerated” and “inertial” means “whatever makes the chosen body free”. The framework requires empirical identification of interactions and a mutually consistent class of free-body trajectories. Removing an influence or improving isolation should make the same inertial description work for many bodies, not merely one fitted trajectory.

Thus the two laws have distinguishable jobs: the first articulates the force-free inertial background; the second quantifies departures caused by impressed forces. A modern axiomatization may build that background into its spacetime structure and then omit a separately numbered first law. That is a legitimate reorganization, but it incorporates the first law's content rather than obtaining all of it from the equation $\mathbf{F}=m\mathbf{a}$. Newton states the laws separately in the opening axioms.^[ref]