Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• The first antiprotons to be observed were produced in the reaction (18.46) (Problem 1), with one of the initial protons at rest. The minimum value of the incident kinetic energy Ki needed to induce the reaction is called the threshold energy for the reaction and is surprisingly large compared to 2mc2. To calculate this threshold energy, note that the minimum of Ki occurs when the final kinetic energy is as small as possible, to maximize the fraction of Ki available to create particles. This requires that all final particles have the same velocity, so that there is no kinetic energy “wasted” in relative motion. Write down the Pythagorean relation for the total energy and momentum of the final particles. (Use the result of Problem 2.) Then use conservation of energy and momentum to rewrite this relation in terms of the energy and momentum, Ei and pi, of the incident projectile. Finally, use the Pythagorean relation for Ei and pi to eliminate pi. Show that Ei = 7mc2 and hence Ki = 6mc2.

Problem 1

•• The first antiprotons to be observed were produced in the reaction

which required a minimum kinetic energy of 6 GeV. Below are listed three reactions that would require less energy, if they were possible. Explain why none of them is possible.

(a)

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(b)

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(c)

Problem 2

• Suppose that n particles all have the same rest mass m and all move with the same velocity u. Prove that their total energy and total momentum satisfy the Pythagorean relation Etot2 = (ptotc)2 + (Mc2)2, appropriate to a single particle of rest mass M = nm.