تکانه

${\displaystyle d\overrightarrow{p}=F.dt}$

${\displaystyle F=\frac{d\overrightarrow{p}}{dt}}$

${\displaystyle \overrightarrow{p}=m\overrightarrow{v}}$

${\displaystyle F=\frac{m.d\overrightarrow{v}}{dt}}$

${\displaystyle a=\frac{\overrightarrow{v}}{dt}}$

${\displaystyle F=m.a}$

${\displaystyle \mathbf{P}=\sum _{i=1}^N{\mathbf{p}}_i={\mathbf{p}}_1+{\mathbf{p}}_2+\cdots +{\mathbf{p}}_N=m_1{\mathbf{v}}_1+m_2{\mathbf{v}}_2+\cdots +m_N{\mathbf{v}}_N}$

${\displaystyle M{\mathbf{r}}_{cm}=\sum _{i=1}^N{m}_i{\mathbf{r}}_i=m_1{\mathbf{r}}_1+m_2{\mathbf{r}}_2+\cdots +m_N{\mathbf{r}}_N}$

${\displaystyle M=\sum _{i=1}^N{m}_i=m_1+m_2+\cdots +m_N}$

${\displaystyle M{\mathbf{V}}_{cm}=\sum _{i=1}^N{m}_i{\mathbf{v}}_i=m_1{\mathbf{v}}_1+m_2{\mathbf{v}}_2+\cdots +m_N{\mathbf{v}}_N}$

${\displaystyle \mathbf{P}={\mathbf{P}}_{cm}=M{\mathbf{V}}_{cm}}$

${\displaystyle \mathbf{F}=\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=\frac{\mathrm{d}{\mathbf{P}}_{cm}}{\mathrm{d}t}}$

${\displaystyle {\mathbf{f}}_i={\mathbf{f}}_i^{ext}+({\mathbf{f}}_{1\to i}^{int}+{\mathbf{f}}_{2\to i}^{int}+\cdots +{\mathbf{f}}_{(i-1)\to i}^{int}+{\mathbf{f}}_{(i+1)\to i}^{int}+\cdots +{\mathbf{f}}_{N\to i}^{int})}$

${\displaystyle \mathbf{F}={\mathbf{F}}_{ext}=\sum _{i=1}^N{\mathbf{f}}_i^{ext}={\mathbf{f}}_1^{ext}+{\mathbf{f}}_2^{ext}+\cdots +{\mathbf{f}}_N^{ext}}$

${\displaystyle {\mathbf{F}}_{ext}=\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=\frac{\mathrm{d}{\mathbf{P}}_{cm}}{\mathrm{d}t}}$

${\displaystyle {\mathbf{F}}_{ext}=0\Rightarrow \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}=0\Rightarrow \mathbf{P}=Const.}$

${\displaystyle {\mathbf{F}}_{ext}=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k}\mathbf{P}=P_x\mathbf{i}+P_y\mathbf{j}+P_z\mathbf{k}}$

${\displaystyle \mathrm{\Delta }{\mathbf{p}}_1+\mathrm{\Delta }{\mathbf{p}}_2=0\Rightarrow \mathrm{\Delta }{\mathbf{p}}_1=-\mathrm{\Delta }{\mathbf{p}}_2}$

${\displaystyle 0=m\mathbf{v}+M\mathbf{V}\Rightarrow \mathbf{V}=-\frac{m}{M}\mathbf{v}}$