15.20: Aceleración
- Page ID
- 130980
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)La Figura XV.33 muestra dos fotogramas de referencia\( \Sigma'\),\( \Sigma\) y, este último moviéndose a velocidad\( v\) con respecto al primero.
Una partícula se mueve con aceleración\( \bf{a'}\) en\( \Sigma'\). (“in\( \Sigma'\)” = “referido al marco de referencia\( \Sigma'\)”.) La velocidad no está necesariamente, por supuesto, en la misma dirección que la aceleración, y supondremos que su velocidad en\( \Sigma'\) es\( \bf{u'}\). Los componentes de aceleración y velocidad en\( \Sigma'\) son\( a'_{x'}, a'_{y'}, u'_{x'}, u'_{y'}\).
¿En qué consiste la aceleración de la partícula\( \Sigma\)? Comenzaremos con el\( x\) -componente.
El\( x\) -componente de su aceleración en\( \Sigma\) viene dado por
\[ a_{x}=\frac{du_{x}}{dt}, \label{15.20.1} \]
donde
\[ u_{x}=\frac{u_{x}'+v}{1+\frac{u_{x}'v}{c^{2}}} \label{15.16.2} \]
y
\[ t=\gamma\left(t'+\frac{vx'}{c^{2}}\right) \label{15.5.19}\tag{15.5.19} \]
Las ecuaciones 15.16.2 y 15.5.19 nos dan
\[ du_{x}=\frac{du_{x}}{du'_{x}}du'_{x}=\frac{du'_{x}}{\gamma^{2}(1+\frac{u'_{x}v}{c^{2}})^{2}} \label{15.20.2} \]
y
\[ dt\ =\frac{\partial t}{\partial t'}dt'+\frac{\partial t}{\partial x'}dx'=\gamma dt'\ +\ \frac{\gamma v}{c^{2}}dx' \label{15.20.3} \]
Al sustituir estos en Ecuación\( \ref{15.20.1}\) y un álgebra muy poco, obtenemos
\[ a_{x}=\frac{a'}{\gamma^{3}(1+\frac{u'_{x}v}{c^{2}})^{3}} \label{15.20.4} \]
El\( y\) -componente de su aceleración en\( \Sigma\) viene dado por
\[ a_{y}=\frac{du_{y}}{dt}, \label{15.20.5} \]
Ya hemos elaborado el denominador\( dt\) (Ecuación\(\ref{15.20.3}\)). Sabemos que
\[ u_{y}=\frac{u'_{y'}}{\gamma(1+\frac{u'_{x'}v}{c^{2}})} \label{15.16.3}\tag{15.16.3} \]
de la cual
\[ du_{y}=\frac{\partial u_{y}}{\partial u'_{x'}}+\frac{\partial u_{y}}{\partial u'_{y'}}\partial u'_{y'}=\frac{1}{\gamma}\left(-\frac{vu'_{y'}}{c^{2}\left(1+\frac{vu'_{x'}}{c^{2}}\right)^{2}}du'_{x'}+\frac{1}{1+\frac{vu'_{x'}}{c^{2}}}du'_{y'}\right). \label{15.20.6} \]
Dividir Ecuación\( \ref{15.20.6}\) por Ecuación\( \ref{15.20.3}\) para obtener
\[ a_{y}=\frac{1}{\gamma^{2}}\left(-\frac{vu'_{y'}}{c^{2}\left(1+\frac{vu'_{x'}}{c^{2}}\right)^{2}}a'_{x'}+\frac{1}{1+\frac{vu'_{x'}}{c^{2}}}a'_{y'}\right). \label{15.20.7} \]