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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 8 and 9}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, March 17, 2003} \\ 
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\centerline{\bf PROBLEM 23}
\centerline{\small{(Problem 12.13 Griffiths)}}
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Sophie Zabar, clairvoyant, cried out in pain at precisely the 
instant that her twin brother, 500 km away, hit his thumb with 
a hammer. A skeptical scientist observed both events (brother's 
accident, Sophie's cry) from an airplaine traveling at 12c/13 
to the right (see Fig.~12.19). Which event occurred first according 
to the scientist? How much earlier was it, in seconds?
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\centerline{\bf PROBLEM 24}
\centerline{\small{(Problem 12.18 Griffiths)}}

\begin{description}
  \item{(a)} Write out the matrix that describes a Galilean
             transformation (Eq.~12.12).
  \item{(b)} Write out the matrix that describing a Lorentz
             transformation along the $y$ axis.
  \item{(c)} Find the matrix describing the Lorentz 
             transformation with velocity $v$ along the
	     $x$ axis followed by a Lorentz
             transformation with velocity $\bar{v}$ along 
	     the $y$ axis. Does it matter in what order
	     the transformations are carried out?
\end{description}
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\centerline{\bf PROBLEM 25}
\centerline{\small{(Problem 12.20 Griffiths)}}

\begin{description}
  \item{(a)} Event $A$ happens at point 
             $(x_{A}\!=\!5,y_{A}\!=\!3,z_{A}\!=\!0)$ 
             and at a time $t_{A}$ given by 
	     $ct_{A}\!=\!15$; event B occurs at
	     $(10,8,0)$ and $ct_{B}\!=\!5$, both in
	     system ${\cal S}$.
	     \begin{description}
               \item{(i)} What is the invariant interval
		          between $A$ and $B$?
               \item{(ii)} Is there an inertial frame in
		           which they occur simultaneously?
			   If so, find its velocity (magnitude
			   and direction) relative to ${\cal S}$.
               \item{(ii)} Is there an inertial frame in
		           which they occur at the same point?
			   If so, find its velocity (magnitude
			   and direction) relative to ${\cal S}$.
             \end{description}
  \item{(b)} Repeat part (a) for $A\!=\!(2,0,0)$, $ct_{A}\!=\!1$;
             and $B\!=\!(5,0,0)$, $ct_{B}\!=\!3$.
\end{description}
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\centerline{\bf PROBLEM 26}
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An electron in its own rest frame has an energy equal to
$E\!=\!m_{0}c^{2}$ and zero linear momentum ${\bf p}\!=\!0$.
Compute the energy and momentum of the electron as measured
by an observer in the {\it ``laboratory''} frame. Note that
in the laboratory frame the electron is moving with constant
linear velocity $v$ along the x-axis; that is,
${\bf v}\!=\!v{\hat{\bf x}}$. Show that while $E$ and $p$
are different in both frames, the quantity $E^{2}-(pc)^{2}$
is invariant.

\medskip
\noindent
{\bf Hint:} Under a Lorentz transformation the energy (momentum)
            transforms as the timelike (spacelike) component of
            a four-vector.
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\centerline{\bf PROBLEM 27}
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Two electrons are moving toward each other. As measured in the
laboratory frame electron 1 has a speed of $v_{1}\!=\!\beta_{1}c$
and electron 2 of $v_{2}\!=\!\beta_{2}c$. What is the speed of
electron 1 in the rest frame of electron 2?
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\centerline{\bf PROBLEM 28}
\medskip

Cosmic rays are very energetically particles (such as protons)
that occasionally collide with the Earth's atmosphere producing
a copious amount of pions---strongly interacting particles with
a finite lifetime. For example, charged pions, which decay via
the weak interaction into muons and neutrinos, have a lifetime
of $\tau\!=\!2.6\times10^{-8}$~s. Compute the total distance
traveled by a pion moving at 99\% of the speed of light as
measured by a {\it ``terrestrial''} observer.

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