M M f VM ex BEFORE AFTER
A photon with wavelength l is incident on a stationary particle with mass M, as shown in Fig. The photon is annihilated while an electron–positron pair is produced. The target particle moves off in the original direction of the photon with speed VM. The electron travels with speed v at angle f with respect to that direction. Owing to momentum conservation, the positron has the same speed as the electron. (a) Using the notation g = (1 - v2/c2)-1/2 and gM = (1 - V 2 M/c2)-1/2, write a relativistic expression for energy conservation in this process. Use the symbol m for the electron mass. (b) Write an analogous expression for momentum conservation. (c) Eliminate the ratio h/l between your energy and momentum equations to derive a relationship between v/c, VM/c, and f in terms of the ratio m/M, keeping g and gM as a useful shorthand notation. (d) Consider the case where VM 66 c, and use a binomial expansion to derive an expression for gM to the first order in VM. Use that result to rewrite your previous result as an expression for VM in terms of c, v, m/M, and f. (e) Are there choices of v and f for which VM = 0? (f) Suppose the target particle is a proton. If the electron and positron remain stationary, so that v = 0, then with what speed does the proton move, in km/s? (g) If the electron and positron each have total energy 5.00 MeV and move with f = 60°, then what is the speed of the proton?
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