15.20: Acceleration
- Page ID
- 8473
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Figure XV.33 shows two references frames, \( \Sigma\) and \( \Sigma'\), the latter moving at speed \( v\) with respect to the former.
A particle is moving with acceleration \( \bf{a'}\) in \( \Sigma'\). (“ in \( \Sigma'\) ” = “referred to the reference frame \( \Sigma'\) ”.) The velocity is not necessarily, of course, in the same direction as the acceleration, and we’ll suppose that its velocity in \( \Sigma'\) is \( \bf{u'}\). The acceleration and velocity components in \( \Sigma'\) are \( a'_{x'}, a'_{y'}, u'_{x'}, u'_{y'}\).
What is the acceleration of the particle in \( \Sigma\)? We shall start with the \( x\)-component.
The \( x\)-component of its acceleration in \( \Sigma\) is given by
\[ a_{x}=\frac{du_{x}}{dt}, \label{15.20.1} \]
where
\[ u_{x}=\frac{u_{x}'+v}{1+\frac{u_{x}'v}{c^{2}}} \label{15.16.2} \]
and
\[ t=\gamma\left(t'+\frac{vx'}{c^{2}}\right) \label{15.5.19}\tag{15.5.19} \]
Equations 15.16.2 and 15.5.19 give us
\[ du_{x}=\frac{du_{x}}{du'_{x}}du'_{x}=\frac{du'_{x}}{\gamma^{2}(1+\frac{u'_{x}v}{c^{2}})^{2}} \label{15.20.2} \]
and
\[ dt\ =\frac{\partial t}{\partial t'}dt'+\frac{\partial t}{\partial x'}dx'=\gamma dt'\ +\ \frac{\gamma v}{c^{2}}dx' \label{15.20.3} \]
On substitution of these into Equation \( \ref{15.20.1}\) and a very little algebra, we obtain
\[ a_{x}=\frac{a'}{\gamma^{3}(1+\frac{u'_{x}v}{c^{2}})^{3}} \label{15.20.4} \]
The \( y\)-component of its acceleration in \( \Sigma\) is given by
\[ a_{y}=\frac{du_{y}}{dt}, \label{15.20.5} \]
We have already worked out the denominator \( dt\)(Equation \(\ref{15.20.3}\)). We know that
\[ u_{y}=\frac{u'_{y'}}{\gamma(1+\frac{u'_{x'}v}{c^{2}})} \label{15.16.3}\tag{15.16.3} \]
from which
\[ du_{y}=\frac{\partial u_{y}}{\partial u'_{x'}}+\frac{\partial u_{y}}{\partial u'_{y'}}\partial u'_{y'}=\frac{1}{\gamma}\left(-\frac{vu'_{y'}}{c^{2}\left(1+\frac{vu'_{x'}}{c^{2}}\right)^{2}}du'_{x'}+\frac{1}{1+\frac{vu'_{x'}}{c^{2}}}du'_{y'}\right). \label{15.20.6} \]
Divide Equation \( \ref{15.20.6}\) by Equation \( \ref{15.20.3}\) to obtain
\[ a_{y}=\frac{1}{\gamma^{2}}\left(-\frac{vu'_{y'}}{c^{2}\left(1+\frac{vu'_{x'}}{c^{2}}\right)^{2}}a'_{x'}+\frac{1}{1+\frac{vu'_{x'}}{c^{2}}}a'_{y'}\right). \label{15.20.7} \]