= Newton's first law
{c}
{title2=$\mathbf F_{\mathrm{net}}=\mathbf0\;\Longrightarrow\;\mathbf v=\mathbf v_0$}
= Law of inertia
{synonym}
In an <inertial frame>, a body with zero net <external force> remains at rest or moves with constant <velocity>. For a <point mass> of constant positive <mass>, write the net <external force> as the <vector> sum of all <external forces> on that body:
$$
\mathbf F_{\mathrm{net}}=\sum_i\mathbf F_i=\mathbf0
\quad\Longrightarrow\quad
\mathbf a=\frac{d\mathbf v}{dt}=\mathbf0,\qquad
\mathbf v(t)=\mathbf v_0,\qquad
\mathbf r(t)=\mathbf r_0+\mathbf v_0(t-t_0).
$$
Here $\mathbf r_0$ and $\mathbf v_0$ are the <position> and <velocity> at <time> $t_0$. The zero net <external force> must hold throughout the interval, not merely at one instant. Both the magnitude and direction of the <velocity> stay fixed: constant <speed> along a curved <trajectory> still involves <acceleration>. Rest is the case $\mathbf v_0=\mathbf0$. For an extended closed system, the analogous statement concerns its <center of mass>; zero net <external force> need not prevent its parts from moving relative to one another.
The short mathematical argument is to use <Newton's second law> with $m>0$, then take the <integral> of $d\mathbf v/dt=\mathbf0$ and of $d\mathbf r/dt=\mathbf v_0$. This gives the two constants fixed by the initial <position> and <velocity>. The physical content also includes the existence of <inertial frames> in which these relations apply; an arbitrary accelerating <reference frame> need not show constant <velocity> for a body with zero actual net <external force>.
The useful question is therefore which <forces> change the motion, rather than which <force> keeps it going. A <free-body diagram> may show several nonzero <forces> whose <vector> sum is zero. Equal opposing <forces> on the same body can balance; a <third-law interaction pair> acts on different bodies and cannot be cancelled within that one body's diagram.
In 1687, <Isaac Newton> stated Law I in \i[Philosophiae Naturalis Principia Mathematica]: a body's rest or uniform straight motion persists unless impressed <forces> compel a change. The following paragraph identifies air resistance and <gravity> as influences on the <trajectory> of a thrown body. The <vector> equations above are a modern expression of the law, not a quotation of his notation.<Philosophiae Naturalis Principia Mathematica by Isaac Newton/Axiomata sive leges motus>[Isaac Newton, Philosophiae Naturalis Principia Mathematica (1687), Axiomata sive leges motus, printed p. 12: Lex I and the following projectile paragraph; normalized transcription]{ref}
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