Newton's Laws of Translational Motion

Axiom: Newton's First Law of Translational Motion

If there is no net force exerted on a point mass, then it moves in a straight-line with constant velocity (or remains still if this velocity is 0).

Definition: Inertial Frame of Reference

An inertial frame of reference is a frame of reference where Newton’s first law of motion applies.

Empirical Law: Newton's Second Law of Translational Motion

In an inertial frame of reference, the acceleration $\mathbf{a}$ of a point mass depends on its rest mass $m$ and the net force $\mathbf{F}$ exerted on it:

$$\mathbf{a}=\frac{\mathbf{F}}{m}$$

Alternatively, the temporal change of the point mass’s momentum is equal to the net force $\mathbf{F}$:

$$\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=\mathbf{F}$$

Theorem: Newton's Second Law of Motion for Physical Systems

In an inertial frame of reference, the acceleration ${\mathbf{a}}_{\text{cm}}$ of the centre of mass of a physical system depends on the vector sum ${\mathbf{F}}_{\text{net}}$ of all forces (both internal and external) exerted on its components and on the total mass $M$ of the system:

$${\mathbf{a}}_{\text{cm}}=\frac{{\mathbf{F}}_{\text{net}}}{M}$$

Alternatively, the temporal change in the total momentum of the system is equal to the vector sum ${\mathbf{F}}_{\text{net}}$ of all forces exerted on its components:

$$\frac{\mathrm{d}}{\mathrm{d}t}\sum \mathbf{p}={\mathbf{F}}_{\text{net}}$$

PROOF

TODO

Empirical Law: Newton's Third Law of Translational Motion

Every interaction between two point masses $A$ and $B$ presents itself as a pair of opposite but equal in magnitude forces ${\mathbf{F}}_{A\text{ on }B}$ and ${\mathbf{F}}_{B\text{ on }A}$:

$${\mathbf{F}}_{B\text{ on }A}=-{\mathbf{F}}_{A\text{ on }B}$$