3.6: Algunas identidades cinemáticas
- Page ID
- 126713
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- Listado de diversas ecuaciones e identidades cinemáticas
Además de las relaciones
\[D(v) = \sqrt{\frac{1+v}{1-v}}\]
y
\[v_c = \frac{v_1 + v_2}{1 + v_1 v_2}\]
las siguientes identidades pueden ser útiles. Si está varado en una isla desierta deberías poder rederivarlos desde cero. No los memorices.
\[v = \frac{D^2 - 1}{D^2 + 1}\]
\[\gamma = \frac{D^{-1} + D}{2}\]
\[v\gamma = \frac{D - D^{-1}}{2}\]
\[D(v)D(-v) = 1\]
\[\eta = \ln D\]
\[v = \tanh \eta\]
\[\gamma = \cosh \eta\]
\[v\gamma = \sinh \eta\]
\[\tanh (x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y}\]
\[D_c = D_1 D_2\]
\[\eta _c = \eta _1 + \eta _2\]
\[v_C \gamma _c = (v_1 + v_2)\gamma _1 \gamma _2\]
Las funciones trigonométricas hiperbólicas se definen de la siguiente manera:
\[\sinh x = \frac{e^x - e^{-x}}{2}\]
\[\cosh x = \frac{e^x + e^{-x}}{2}\]
\[\tanh x = \frac{\sinh x}{\cosh x}\]
Sus inversos están integrados en algunas calculadoras y programas informáticos, pero también se pueden calcular usando las siguientes relaciones:
\[\sinh^{-1}x = \ln \left ( x + \sqrt{x^2 + 1} \right )\]
\[\cosh^{-1}x = \ln \left ( x + \sqrt{x^2 - 1} \right )\]
\[\tanh^{-1}x = \frac{1}{2}\ln \left ( \frac{1 + x}{1 - x} \right )\]
Sus derivados son, respectivamente,\(\left ( x^2 + 1 \right )^{-1/2}\),\(\left ( x^2 - 1 \right )^{-1/2}\) y\(\left ( 1 - x^2 \right )^{-1}\).