= Acceleration
{title2=$\mathbf a=d\mathbf v/dt$}
{wiki}
<Acceleration> is the <time> <derivative> of <velocity>, equivalently the second <derivative> of <position>:
$$
\mathbf a(t)=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}.
$$
It is a <vector> with <SI> unit $\mathrm{m/s^2}$. A change in direction produces <acceleration> even if <speed> stays constant. For example, a car rounding a bend at steady <speed> needs an inward resultant <force> by <Newton's second law>. Thus <acceleration> does not mean only speeding up. When $v>0$, the <dot product> gives $dv/dt=\mathbf v\cdot\mathbf a/v$: a perpendicular <acceleration> changes direction without changing <speed> at that instant.
Take counterclockwise <circular motion> about the origin with <radius> $R=2\,\mathrm m$ and constant <angular speed> $\omega=0.5\,\mathrm{rad/s}$. The <position>, <velocity> and <acceleration> are
$$
\begin{aligned}
\mathbf r(t)&=(R\cos(\omega t),R\sin(\omega t)),\\
\mathbf v(t)&=(-R\omega\sin(\omega t),R\omega\cos(\omega t)),\\
\mathbf a(t)&=(-R\omega^2\cos(\omega t),-R\omega^2\sin(\omega t))=-\omega^2\mathbf r(t).
\end{aligned}
$$
The <speed> is $R\omega=1\,\mathrm{m/s}$ throughout, while the inward <centripetal acceleration> has magnitude $R\omega^2=v^2/R=0.5\,\mathrm{m/s^2}$. At the four cardinal points, reached at $t=0,\pi,2\pi,3\pi\,\mathrm s$, the <velocities> in $\mathrm{m/s}$ are $(0,1),(-1,0),(0,-1),(1,0)$; the <accelerations> in $\mathrm{m/s^2}$ are $(-0.5,0),(0,-0.5),(0.5,0),(0,0.5)$. The <vectors> turn together, remaining perpendicular. https://openstax.org/books/university-physics-volume-1/pages/4-4-uniform-and-nonuniform-circular-motion[Ling, Sanny and Moebs, section 4.4: uniform circular motion]{ref}
\Image[/constant-speed-circular-acceleration.png]
{title=Constant speed with changing velocity and inward acceleration at four points of a circle}
The proof outline is just two applications of the <chain rule> to $\mathbf r(t)$; $\sin^2(\omega t)+\cos^2(\omega t)=1$ then fixes both magnitudes. The usefulness is geometric: turning requires a <force> even when a <speed> reading remains steady, and doubling <speed> at fixed <radius> quadruples the required <centripetal acceleration>. The same mechanism appears in a <circular orbit>.
Historically, Newton's \i[Principia], Book I, Section II, Proposition IV, relates the inward <forces> of uniform <circular motion> to the squared arcs traversed in equal <times>, divided by their <radii>. In modern notation, its dependence on <speed> and <radius> is $v^2/R$; Newton presents a geometric argument rather than the coordinate <derivatives> above. https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Book_1/Section_2[Newton, Principia, Book I, Proposition IV; Motte's 1729 translation]{ref}
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