In class we, briefly, discussed the example of the kinetic energy of a beam particle colliding with a target particle being transformed into the masses of new particles in the final state. some of these final state particles could just be the original projectile and target, but they don't have to be. assume that a beam particle b, with total relativistic energy eb collides with a target particle t' that is at rest, so only has total relativistic energy mr. new particles pı, p2, are produced in the final state. we can write the reaction as: (a) show that the minimum energy emin needed is: 2 2 2 emin = 2mt where m = m1 + m2 + + mn, the sum of the rest masses of all the final state particles. this minimum energy, egin, is known as the threshold for producing the final state p p2 .. p (b) calculate the minimum π-(139) momentum necessary in a beam of projectiles for the reaction to occur. experiments that actually do this, typically fire the pion beam (with the momentum you will determine) into a liquid hydrogen target where the protons are at rest. (c) in another experiment one could attempt to produce the lightest t(1s) meson by firing a beam of positrons (e+) on a target that contains (effectively) stationary electrons, e . these electrons could also be found in a liquid hydrogen target, like the one above. what energy would the positrons need to successfully create t(1s) mesons? for a bonus point why do we not have to worry about the momentum of the electrons in the hydrogen atoms in the target? (d) consider, instead, the case where one made a beam of both electrons and positrons with equal energies, but traveling in opposite directions. how much energy (or momentum) would the beams have to have to create a r(15) meson in this case?