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\begin{center}
{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf Second Midterm Exam}\\%
\vspace {1mm}{April 7, 2003} \\ 
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\centerline{\bf PROBLEM 1}
\noindent
Write the two basic postulates of the Special Theory of Relativity.
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\centerline{\bf PROBLEM 2}
\noindent 
A bullet is fired from a train at a velocity $v_{b}$ (to the right)
and rapidity $\xi_{b}$. The train itself is moving relative to a
stationary platform at a velocity $v_{t}$ (also to the right) and
rapidity $\xi_{t}$. What are the velocity and rapidity of the bullet 
as measured by an observer standing in the platform.
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\centerline{\bf PROBLEM 3}
\noindent
Consider an event $A$ that happens at a space-time point
$(x_{A}\!=\!5,y_{A}\!=\!3,z_{A}\!=\!0)$ and at a time 
$t_{A}$ given by $ct_{A}\!=\!15$, and an event B that occurs 
at $(10,8,5)$ 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? Answer yes or no and explain your
	      answer but do {\bf NOT} find the frame even if one 
	      exists.
 \item{(iii)} Is there an inertial frame in which they occur 
              at the same point? Answer yes or no and explain
	      your answer but do {\bf NOT} find the frame even
	      if one exists.
\end{description}
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\centerline{\bf PROBLEM 4}
\noindent
Consider a massless $(m\!=\!0)$ photon traveling through
empty space with a zero linear momentum $({\bf p}\!=\!{\bf 0})$.
\begin{description}
 \item{(i)}  What is the magnitude of the velocity of the photon?
 \item{(ii)} What is the energy of the photon? 
\end{description}
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\centerline{\bf PROBLEM 5}
\noindent
Consider an electron of charge $e$ and mass $m$ traveling
unperturbed through space with linear momentum ${\bf p}$
and energy $E\!=\!\sqrt{({\bf p}c)^{2}\!+\!(mc^{2})^2}$.
How do these four quantities (charge, mass, linear momentum, 
and energy) transform under a Lorentz transformation?
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\centerline{\bf PROBLEM 6}
\noindent 
Consider the head-on collision of an electron and a positron
(identical in all respects except for their opposite charges)
each with energy $E\!=\!1$~GeV. In the center-of-momentum frame 
their respective four-momenta are given by:
$$
  p_{e^{+}}^{\mu}=(E,+pc)\;; \quad
  p_{e^{-}}^{\mu}=(E,-pc)\;,
$$
where $pc\!=\!\sqrt{E^{2}\!-\!(mc^{2})^2}$. The Mandelstam 
variable $s$ for this collision is given by
$$
  s=(p_{e^{+}}+p_{e^{-}})\cdot(p_{e^{+}}+p_{e^{-}})
   \equiv(p_{e^{+}}+p_{e^{-}})^{\mu}(p_{e^{+}}+p_{e^{-}})_{\mu}
   =4E^{2}=4~{\rm GeV}^{2}\;.
$$
\begin{description}
 \item{(i)}  What is the numerical value of $s$ in the 
             rest-frame of the positron?
 \item{(ii)} What is the numerical value of $s$ in the 
             rest-frame of the electron?
\end{description}
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