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Physics LibreTexts

3.6: Some kinematic identities

Skills to Develop

  • 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 \(\left ( 1 - x^2 \right )^{-1}\).

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