3.6: Some kinematic identities
- Page ID
- 13034
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\ket}[1]{\left| #1 \right>}\)
\(\newcommand{\bra}[1]{\left< #1 \right|}\)
\(\newcommand{\braket}[2]{\left< #1 \vphantom{#2} \right| \left. #2 \vphantom{#1} \right>}\)
\(\newcommand{\braopket}[3]{\left< #1 \vphantom{#2}\vphantom{#3} \right| #2 \vphantom{#1}\vphantom{#3} \left| #3 \vphantom{#1}\vphantom{#2} \right>}\)
\(\newcommand{\qmvec}[1]{\mathbf{\vec{#1}}}\)
\(\newcommand{\op}[1]{\hat{\mathbf{#1}}}\)
\(\newcommand{\expect}[1]{\langle #1 \rangle}\)
\(\newcommand{\dfn}[1]{\emph{\textbf{#1}}}\)
Learning Objectives
- List of various kinematics equations and identities
In addition to the relations
\[D(v) = \sqrt{\frac{1+v}{1-v}}\]
and
\[v_c = \frac{v_1 + v_2}{1 + v_1 v_2}\]
the following identities can be handy. If stranded on a desert island you should be able to rederive them from scratch. Don’t memorize them.
\[v = \frac{D^2 - 1}{D^2 + 1}\]
\[\gamma = \frac{D^{-1} + D}{2}\]
\[v\gamma = \frac{D - D^{-1}}{2}\]
\[D(v)D(-v) = 1\]
\[\eta = \ln D\]
\[v = \tanh \eta\]
\[\gamma = \cosh \eta\]
\[v\gamma = \sinh \eta\]
\[\tanh (x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}\]
\[D_c = D_1 D_2\]
\[\eta _c = \eta _1 + \eta _2\]
\[v_C \gamma _c = (v_1 + v_2)\gamma _1 \gamma _2\]
The hyperbolic trig functions are defined as follows:
\[\sinh x = \frac{e^x - e^{-x}}{2}\]
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
\[\tanh x = \frac{\sinh x}{\cosh x}\]
Their inverses are built in to some calculators and computer software, but they can also be calculated using the following relations:
\[\sinh^{-1}x = \ln \left ( x + \sqrt{x^2 + 1} \right )\]
\[\cosh^{-1}x = \ln \left ( x + \sqrt{x^2 - 1} \right )\]
\[\tanh^{-1}x = \frac{1}{2}\ln \left ( \frac{1 + x}{1 - x} \right )\]
Their derivatives are, respectively, \(\left ( x^2 + 1 \right )^{-1/2}\), \(\left ( x^2 - 1 \right )^{-1/2}\) and \(\left ( 1 - x^2 \right )^{-1}\).