3.6: Some kinematic identities
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 \(\left ( 1 - x^2 \right )^{-1}\).