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{\bf ADVANCED DYNAMICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 10}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, March 24, 2003} \\ 
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\centerline{\bf PROBLEM 29}
\centerline{({\small Griffiths Problem 12.34})}

In the past, most experiments in particle physics involved
stationary targets: one particle (usually a proton or an 
electron) was accelerated to high energy E, and collided
with a target particle at rest. Far higher {\it relative}
energies are obtainable (with the same accelerator) if you
accelerate {\it both} particles to energy $E$, and fire them
at each other. {\it Classically}, the energy $\bar{E}$ of
one particle relative to the other is just $4E$---not much
of a gain (only a factor of 4). But {\it relativistically}
the gain can be enormous. Assuming that the two particles
have the same mass ($m$), show that
$$
	\bar{E}=\frac{2E^{2}}{mc^{2}}-mc^{2}\;.
$$
Suppose that you use protons ($mc^{2}\!\approx\!1$~GeV)
with $E\!=\!30$~GeV. What $\bar{E}$ do you get? What
multiple of $E$ does this amount to? [Because of the
relativistic enhancement, most modern elementary particle
experiments involve {\bf colliding beams}, instead
of fixed targets.]
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\centerline{\bf PROBLEM 30}
\centerline{({\small Griffiths Problem 12.35})}

In a {\bf pair annihilation} experiment, an electron
(mass $m$) with momentum $p_{e}$ hits a positron (same
mass but opposite charge) at rest. They annihilate
producing two photons. (Why couldn't they produce just
{\it one} photon?) If one of the photons emerges at
$60^{\circ}$ to the incident electron direction, what
is its energy?
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\centerline{\bf PROBLEM 31}
\centerline{({\small Griffiths Problem 12.58})}

Calculate the {\bf threshold} (minimum) momentum that a
pion must have (in the laboratory frame) in order for 
the process $\pi + p \rightarrow K + \Sigma$ to occur. 
Note that in the laboratory frame the proton is initially 
at rest. Use the following {\it approximate} values for
the masses of all the particles:
\begin{eqnarray}
   & M_{\pi}c^{2}=150~{\rm MeV} \,,
   & M_{K}c^{2}=500~{\rm MeV} \;;\nonumber \\
   & M_{p}c^{2}=900~{\rm MeV} \,,
   & M_{\Sigma}c^{2}=1200~{\rm MeV}\;. \nonumber 
\end{eqnarray}
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\noindent
{\bf Hint:} Formulate the threshold condition in the 
	    center-of-momentum frame.
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