= Newton's second law
{c}
{title2=$\mathbf F_{\mathrm{net}}=m\mathbf a$}
In an <inertial frame>, a <point mass> of constant <mass> $m>0$ obeys
$$
\mathbf F_{\mathrm{net}}=\sum_i\mathbf F_i
=\frac{d\mathbf p}{dt}=m\mathbf a,
\qquad \mathbf p=m\mathbf v,
\qquad \mathbf a=\frac{d\mathbf v}{dt}.
$$
The net <force> is the sum of all <forces> acting on that body; <momentum>, <velocity> and <acceleration> refer to the same <reference frame>. A <force> changes <velocity>, rather than sustaining it: zero net <force> implies constant <velocity>, including a nonzero one. Doubling the net <force> at fixed <mass> doubles <acceleration>; doubling <mass> at fixed net <force> halves it. https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/mit8_01scs22_chapter7.pdf[MIT mechanics, Chapter 7, section 7.3]{ref}
For example, an otherwise balanced block of <mass> $0.5\,\mathrm{kg}$ receives a constant horizontal net <force> $2\,\mathrm N$ for $0.5\,\mathrm s$. Its <acceleration> is $4\,\mathrm{m/s^2}$, so its <velocity> changes by $2\,\mathrm{m/s}$. Equivalently, the <impulse> is $F\Delta t=1\,\mathrm{N\,s}$, and $\Delta v=\Delta p/m$. Starting from rest, its <displacement> during the pulse is $a(\Delta t)^2/2=0.5\,\mathrm m$. After the pulse, zero net <force> leaves it moving at constant <velocity>.
Applying the law therefore requires both a specification of the net <force> and <initial conditions>. In the <harmonic oscillator> example, the <force> is $-kx$, so <Newton's second law> becomes a <differential equation> for <position>. In an accelerating <reference frame>, the measured relative <acceleration> includes a contribution from the frame's own motion; the <equation of motion in a rotating frame> illustrates the extra terms required.
Constant <mass> is essential to the displayed $m\mathbf a$ form. A <variable-mass system> exchanging matter also exchanges <momentum>; its balance includes the entering or leaving matter's <velocity>. Simply differentiating $m\mathbf v$ and equating it to external <force> on that open body can omit this flux. The existing <variable-mass system> equation specifies the needed correction.
In the 1687 \i[Principia], <Newton's second law> was stated in terms of change of motion proportional to impressed <force>, directed along that <force>. Its original statement is verbal; the <derivative> equation above is modern notation. <Philosophiae Naturalis Principia Mathematica by Isaac Newton/Axiomata sive leges motus>[Philosophiae Naturalis Principia Mathematica by Isaac Newton; Newton, Axiomata sive leges motus (1687), Lex II]{ref}
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