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