Two protons (mass = 1.67 x 10 -27 kg) withequal speeds in opposite directions co

Question

Two protons (mass = 1.67 x 10-27 kg) withequal speeds in opposite directions collide head-on, producing aparticle called a pi meson (0, mass = 2.40 x10-28 kg). These reactions are how the Large HadronCollider is producing new particles. (a) If all the particles are at rest after the collision, findthe initial kinetic energy and initial speed of each proton. (b) What is the initial momentum of one of the protons beforethe collision? (as measured from an observer in the lab frame atrest) (c) How is the rest energy of the pi meson related to thetotal initial kinetic energy of the protons? Two protons (mass = 1.67 x 10-27 kg) withequal speeds in opposite directions collide head-on, producing aparticle called a pi meson (0, mass = 2.40 x10-28 kg). These reactions are how the Large HadronCollider is producing new particles. (a) If all the particles are at rest after the collision, findthe initial kinetic energy and initial speed of each proton. (b) What is the initial momentum of one of the protons beforethe collision? (as measured from an observer in the lab frame atrest) (c) How is the rest energy of the pi meson related to thetotal initial kinetic energy of the protons?

Explanation / Answer

m = mp = 1.67 x 10-27 kg, m' =m = 2.40 x 10-28 kg a) find the initial kinetic energy of each proton K energy conservation: 2(mc2 + K) = 2mc2 + m'c2 so K = m'c2/2 = 1.08*10-11 J K = ( - 1)mc2 so = 1 + K/(mc2) = 1 + m'/(2m) = 1.072 (1 - v2/c2) = 1/ 1 - v2/c2 = 1/2 = 0.87 v/c = 0.13 v = 3.89*107 m/s b) p = mv = 6.96*10-20 kg-m/s c) m'c2 = 2K so the rest energy of the pi meson and the total initial kineticenergy of the protons are equal.

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