# 加速度

a

## 簡述

### 直線運動中的平均加速度和瞬時加速度

${\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {\mathrm {d} x}{\mathrm {d} t}}}$

${\displaystyle {\bar {a}}={\frac {v(t+\Delta t)-v(t)}{\Delta t}}}$

${\displaystyle a(t)=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} x}{\mathrm {d} t^{2}}}}$

${\displaystyle v(t_{1})=\int _{t_{0}}^{t_{1}}a(t)\mathrm {d} t+v(t_{0})}$

### 曲線運動中的加速度

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

${\displaystyle \Delta t}$足夠小時，可以將那一小段曲線運動（稱作元弧）近似看作直線運動或圓周運動[1]:30

### 伽利略變換

${\displaystyle \mathbf {a} =\mathbf {a} '+\mathbf {a} _{\text{rel}}}$

### 牛頓第二定律

${\displaystyle \mathbf {F} =m\mathbf {a} }$

### 慣性力

${\displaystyle \sum _{i}\mathbf {F} _{i}=m\mathbf {a} }$

${\displaystyle \sum _{i}\mathbf {F} _{i}-m\mathbf {a} =0}$

${\displaystyle \sum _{i}\mathbf {F} +\mathbf {I} =0}$

### 角加速度

${\displaystyle \alpha (t)={\frac {\mathrm {d} \omega (t)}{\mathrm {d} t}}={\frac {\mathrm {d^{2}} \theta (t)}{\mathrm {d} t^{2}}}}$

## 加速度的分解

### 按坐標系分解

#### 平面直角坐標系

${\displaystyle \mathbf {a} (t)=a_{x}(t)\mathbf {i} +a_{y}(t)\mathbf {j} }$

#### 極坐標系

${\displaystyle x=r\cos \theta }$
${\displaystyle y=r\sin \theta }$

${\displaystyle \mathbf {r} =r\mathbf {e} _{r}}$
${\displaystyle \mathbf {v} =r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {dr}{\mathrm {d} t}}\mathbf {e} _{r}}$
${\displaystyle \mathbf {a} =\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }}$
##### 極坐標系分解的推導

${\displaystyle \mathrm {d} \mathbf {e} _{r}=\mathrm {d} \theta \mathbf {e} _{\theta }}$
${\displaystyle \mathrm {d} \mathbf {e} _{\theta }=-\mathrm {d} \theta \mathbf {e} _{r}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}(a\ b)=a\ {\frac {\mathrm {d} b}{\mathrm {d} t}}+b\ {\frac {\mathrm {d} a}{\mathrm {d} t}}}$

{\displaystyle {\begin{aligned}\mathbf {v} &={\frac {\mathrm {d} }{\mathrm {d} t}}(r\mathbf {e} _{r})\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\\\end{aligned}}}
{\displaystyle {\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\&={\frac {\mathrm {d} }{\mathrm {d} t}}\left(r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\right)+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\mathrm {d} r}{\mathrm {d} t}}\mathbf {e} _{r}\right)\\&=r{\frac {\mathrm {d} \theta }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{\theta }}{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {e} _{r}}{\mathrm {d} t}}\\&=-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\mathbf {e} _{r}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }+{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}\mathbf {e} _{r}+{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {e} _{\theta }\\&=\left[{\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]\mathbf {e} _{r}+\left(2{\frac {\mathrm {d} r}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+r{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right)\mathbf {e} _{\theta }\\\end{aligned}}}

#### 按自然坐標系分解

${\displaystyle a_{\text{t}}={\frac {\mathrm {d} v}{\mathrm {d} t}}}$
${\displaystyle a_{\text{n}}={\frac {v^{2}}{\rho }}}$

### 按功能分解

#### 等速率圓周運動 向心加速度

${\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}$

${\displaystyle \mathbf {a} _{\text{n}}=\mathbf {\omega } \times (\mathbf {\omega } \times \mathbf {r} )}$

##### 向心加速度的推導

${\displaystyle a_{\text{n}}=v{\frac {\mathrm {d} \beta }{\mathrm {d} t}}=v\omega }$

${\displaystyle v=\omega r}$

${\displaystyle a_{\text{n}}=\omega ^{2}r={\frac {v^{2}}{r}}}$

#### 科里奧利效應

${\displaystyle \mathbf {a} _{\text{cor}}=-2{\boldsymbol {\Omega }}\times \mathbf {v} }$
${\displaystyle \mathbf {F} _{\text{cor}}=-2m{\boldsymbol {\Omega }}\times \mathbf {v} }$

#### 歐拉力

${\displaystyle \mathbf {F} _{\text{Euler}}=m\mathbf {a} _{\text{Euler}}}$

${\displaystyle \mathbf {a} _{\text{Euler}}=-{\frac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} }$

${\displaystyle \mathbf {F} _{\text{Euler}}=-m{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {r} }$

## 幾種特殊的運動

### 勻變速直線運動

${\displaystyle v(t)=v_{0}+at\,.}$
{\displaystyle {\begin{aligned}s(t)&=v_{0}t+{\frac {1}{2}}at^{2}\\&={\frac {v(t)+v_{0}}{2}}t\\\end{aligned}}}

${\displaystyle a={\frac {v(t)^{2}-v_{0}^{2}}{2s(t)}}}$

### 簡諧運動

${\displaystyle x=A\cos(\omega t+\phi _{0})\,.}$

${\displaystyle v=-A\omega \sin(\omega t+\phi _{0})\,}$
${\displaystyle a=-A\omega ^{2}\cos(\omega t+\phi _{0})}$

${\displaystyle a=-x\omega ^{2}\,}$
${\displaystyle A^{2}=\left({\frac {a}{\omega ^{2}}}\right)^{2}+\left({\frac {v}{\omega }}\right)^{2}\,}$

## 加速度的應用

### 狹義相對論

${\displaystyle {\begin{cases}x=\gamma (x'+vt)\\y=y'\\z=z'\\t=\gamma \left(t'+{\cfrac {v}{c^{2}}}x\right)\\\end{cases}}}$

${\displaystyle {\begin{cases}{\cfrac {\mathrm {d} x}{\mathrm {d} t}}=u_{x}\\{\cfrac {\mathrm {d} u_{x}}{\mathrm {d} t}}=a_{x}\end{cases}}}$

y、z方向的定義式與之類似。綜合該定義式，利用坐標轉換的t部分，將坐標轉換的x、y、z連續兩次進行一階求導[2]:501

${\displaystyle {\begin{cases}a_{x}={\cfrac {a'_{x}}{\gamma ^{3}\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\\a_{y}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{y}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{y}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\a_{z}={\cfrac {1}{\gamma ^{2}}}\left[{\cfrac {a'_{z}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{2}}}-{\cfrac {{\frac {vu'_{z}}{c^{2}}}a'_{x}}{\left(1+{\frac {vu'_{x}}{c^{2}}}\right)^{3}}}\right]\\\end{cases}}}$

## 參考文獻

1. 趙凱華，羅蔚茵. 《新概念物理教程·力学（第二版）》. 北京: 高等教育出版社. 2004. ISBN 7-04-015201-0.
2. 鄭永令，賈起民，方小敏. 《力学（第二版）》. 北京: 高等教育出版社. 2002. ISBN 978-7-04-011084-5.
3. Beer, Ferdinand; Johnston, Jr., E. Russ, Vector Mechanics for Engineers:dynamics 7th: pp. 699, 2004, ISBN 978-0-072-93079-5
4. Lanczos, Cornelius, The Variational Principles of Mechanics, Dovers Publications, Inc, 1970, ISBN 978-0-486-65067-8
5. ^ 存档副本. [2013-02-19]. （原始內容存檔於2013-05-18）.
6. ^ 黃沛天，馬善鈞，徐學翔，胡利雲．變加速動力學縱橫，2010年7月5日查詢
7. ^ Sprott JC. Some simple chaotic jerk functions (PDF). Am J Phys. 1997, 65 (6): 537–43 [2009-09-28]. Bibcode:1997AmJPh..65..537S. doi:10.1119/1.18585. （原始內容 (PDF)存檔於2010-06-13）.
8. ^ Baum, Steven K，1997年1月20日，The Glossary: Cn-Cz.. Glossary of Oceanography and the Related Geosciences with References. Texas A&M University，於2006年11月29日。（英文）
9. ^ Tropical cyclone: Tropical cyclone tracks.. Encyclopædia Britannica. 2008-02-25 [2009-05-07]. （原始內容存檔於2012-06-22）.
10. ^ NOAA FAQ: How much energy does a hurricane release?. National Oceanic & Atmospheric Administration. August 2001 [2009-06-30]. （原始內容存檔於2017-11-02）.
11. ^ 豎直向下，又稱鉛直向下，被定義為重力加速度的方向。但其具體方向因重力加速度的兩種定義不同而異，分別為指向地心與緯度有關，參見萬有引力#兩者的微妙差別
12. ^ g在不同地區稍有不同，並且g有兩種不同的定義（見上一條注釋）。一般需要更精確的計算中g可近似的取作標準重力加速度，即g=gn=9.80665 ms-2，這個值是已經包括了和地球自轉的向心力的。該數值來自氣象港，http://qxg.com.cn/n/?cid=44&nid=764&fc=nd，2010年5月18日查阅。頁面存檔備份，存於網際網路檔案館
13. ^ 這裡並沒有用到準確的物理術語。準確地說，是輻射的能流密度與粒子加速度的平方成正比。趙凱華，陳熙謀. 《新概念物理教程·电磁学》. 北京: 高等教育出版社. 2003: pp.417–419. ISBN 7-04-011693-6.
14. ^ French, Anthony, Special Relativity (Mit Introductory Physics Series), United States of America: W. W. Norton: pp. 153–154, 1968, ISBN 978-0748764228 （英語）