# 基礎力學/運動的描述

## 位移與路程

${\displaystyle \mathbf {\Delta r} =\mathbf {r} (t_{0}+\Delta t)-\mathbf {r} (t_{0})}$

{\displaystyle {\begin{aligned}\mathbf {\Delta r} &=\mathbf {r} (t_{0}+\Delta t)-\mathbf {r} (t_{0})\\&=[x(t_{0}+\Delta t)\mathbf {i} +y(t_{0}+\Delta t)\mathbf {j} +z(t_{0}+\Delta t)\mathbf {k} ]-[x(t_{0})\mathbf {i} +y(t_{0})\mathbf {j} +z(t_{0})\mathbf {k} ]\\&=[x(t_{0}+\Delta t)-x(t_{0})]\mathbf {i} +[y(t_{0}+\Delta t)-y(t_{0})]\mathbf {j} +[z(t_{0}+\Delta t)-z(t_{0})]\mathbf {k} \\&=\Delta x\mathbf {i} +\Delta y\mathbf {j} +\Delta z\mathbf {k} \end{aligned}}}

## 速度與加速度

《水經注》中描述長江在三峽水流湍急時用「或王命急宣，有時朝發白帝，暮到江陵。」[1]來形容。文中的以船航行時間之短（僅在朝暮之間），及航行起末位置間隔長（從白帝到江陵）來突出江水流速之快。在運動學中，我們以位移與時間的比值來定義平均速度，記作${\displaystyle {\overline {\mathbf {v} }}}$

${\displaystyle {\overline {\mathbf {v} }}={\frac {\mathbf {\Delta r} }{\Delta t}}={\frac {\mathbf {r} (t_{0}+\Delta t)-\mathbf {r} (t_{0})}{\Delta t}}}$

${\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\overline {\mathbf {v} }}=\lim _{\Delta t\to 0}{\frac {\mathbf {\Delta r} }{\Delta t}}}$

${\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}(t)={\frac {d\mathbf {r} }{dt}}}$

${\displaystyle {\overline {\mathbf {a} }}={\frac {\mathbf {\Delta v} }{\Delta t}}={\frac {\mathbf {v} (t_{0}+\Delta t)-\mathbf {v} (t_{0})}{\Delta t}}}$

${\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\overline {\mathbf {a} }}=\lim _{\Delta t\to 0}{\frac {\mathbf {\Delta v} }{\Delta t}}={\frac {d\mathbf {v} }{dt}}}$

${\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}(t)={\frac {d\mathbf {r} }{dt}}}$

${\displaystyle \mathbf {a} ={\ddot {\mathbf {r} }}(t)={\frac {d^{2}\mathbf {r} }{dt^{2}}}}$

{\displaystyle {\begin{alignedat}{2}\mathbf {v} &=v_{x}\mathbf {i} +v_{y}\mathbf {j} +v_{z}\mathbf {k} \\\mathbf {a} &=a_{x}\mathbf {i} +a_{y}\mathbf {j} +a_{z}\mathbf {k} \end{alignedat}}}

{\displaystyle {\begin{alignedat}{2}\mathbf {v} &={\frac {d\mathbf {r} }{dt}}={\frac {dx}{dt}}\mathbf {i} +{\frac {dy}{dt}}\mathbf {j} +{\frac {dz}{dt}}\mathbf {k} \\\mathbf {a} &={\frac {d\mathbf {v} }{dt}}={\frac {dv_{x}}{dt}}\mathbf {i} +{\frac {dv_{y}}{dt}}\mathbf {j} +{\frac {dv_{z}}{dt}}\mathbf {k} \end{alignedat}}}

{\displaystyle {\begin{alignedat}{2}v_{x}&={\frac {dx}{dt}},&v_{y}&={\frac {dy}{dt}},&v_{z}&={\frac {dz}{dt}}.\\a_{x}&={\frac {dv_{x}}{dt}}={\frac {d^{2}x}{dt^{2}}},\quad &a_{y}&={\frac {dv_{y}}{dt}}={\frac {d^{2}y}{dt^{2}}},\quad &a_{z}&={\frac {dv_{z}}{dt}}={\frac {d^{2}z}{dt^{2}}}.\end{alignedat}}}

## 注釋

1. 酈道元.水經注·注水.北魏
2. 在不引起混淆的前提下，我們約定，本書中出現的「速度」都是指的「瞬時速度」。
3. 在不引起混淆的前提下，我們約定，本書中出現的「加速度」都是指的「瞬時加速度」。
4. ${\displaystyle \mathbf {r} =\mathbf {r} (t)=x(t)\cdot \mathbf {i} +y(t)\cdot \mathbf {j} +z(t)\cdot \mathbf {k} }$