A beam of mesons of mass and kinetic energy travels along the axis of the laboratory…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Particle Physics & Quantum Fields · Particles, interactions and experiment

Relativity and Gravitation, Ch Relativistic momentum, energy and collisions

Problem

A beam of

\pi^-

mesons of mass

m_\pi = 139.570\ \mathrm{MeV}/c^2

and kinetic energy

K = 1.000\ \mathrm{GeV} = 1000.0\ \mathrm{MeV}

travels along the

+x

axis of the laboratory and strikes protons at rest in a target,

m_p = 938.272\ \mathrm{MeV}/c^2

. Consider one pion and one target proton as an isolated two-particle system just before the collision. Take

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

. ((a)) Find the invariant mass

M_{\rm inv}

of this system, in

\mathrm{MeV}/c^2

. ((b)) Find the speed of the centre-of-momentum frame relative to the laboratory, as a multiple of

c

. ((c)) The threshold condition says the total mass of the particles emerging from the collision can be at most

M_{\rm inv}

. By how much, in

\mathrm{MeV}/c^2

, does this exceed the total mass of the two incoming particles?

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