${\displaystyle {\vec {a}}\;{\overset {def}{=}}\lim _{\Delta t\to 0}{\frac {\Delta v}{\Delta t}}={\frac {d{\vec {v}}}{dt}}={\dot {\vec {v}}}.}$

(1)

${\displaystyle {\vec {a}}={\frac {d{\vec {v}}}{dt}}={\frac {d}{dt}}\left({\frac {d{\vec {r}}}{dt}}\right)={\frac {d^{2}{\vec {r}}}{dt^{2}}}={\ddot {\vec {r}}}.}$

(2)

${\displaystyle {\begin{cases}a_{x}={\dot {v_{x}}}={\ddot {x}},\\a_{y}={\dot {v_{y}}}={\ddot {y}}\\a_{z}={\dot {v_{z}}}={\ddot {z}}\end{cases}},}$

(3)

${\displaystyle [a]=LT^{-2},}$.

(4)

${\displaystyle {\frac {dv}{dt}}=a_{t}=const.}$

(5)

${\displaystyle \int dv=\int a_{t}dt\;\Rightarrow \;v=v_{0}+a_{t}(t-t_{0})}$

(6)

${\displaystyle \int ds=\int v\cdot dt\;\Rightarrow \;s=s_{0}+v_{0}(t-t_{0})+{\frac {1}{2}}a_{t}(t-t_{0})^{2}.}$

(7)

(8)

${\displaystyle s=s_{0}+{\frac {v_{0}+v}{2}}\cdot (t-t_{0})=s_{0}+{\bar {v}}(t-t_{0}),\;\;{\bar {v}}={\frac {v_{0}+v}{2}}.}$

(9)