A new theory predicts that a chargino x¯ of mass m(x¯¯) = 620 GeV decays into a neutralino x with a mass of m(x) = 171 GeV and a W boson with a mass of m(W) 80.4 GeV. a) Calculate the energy of the neutralino in the rest frame of the chargino.

A muon has a rest mass energy of 105.7 MeV, and it decays into an electron and a massless particle. (a) If all the lost mass is converted into the electron's kinetic energy, find for the electron. (b) What is the electron's velocity?

Show that for a particle is invariant under Lorentz transformations.

(a) Beta decay is nuclear decay in which an electron is emitted. If the electron is given 0.750 MeV of kinetic energy, what is its velocity? (b) Comment on how the high velocity is consistent with the kinetic energy as it compares to the rest mass energy of the electron.

A neutral meson is a particle that can be created by accelerator beams. If one such particle lives as measured in the laboratory, and when at rest relative to an observer, what is its velocity relative to the laboratory?

The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.

Say that the mean lifetime of a π+ meson in its own rest frame is τ = 2.6 x 10-8 s. A pion of this lifetime is created at an altitude 100 km in the atmosphere by the collision of an incoming cosmic·ray proton with an atmospheric nucleus, and it has lifetime τ. How fast would this π+ meson need to move in order to reach the ground before decaying? Express the velocity in the form v/c = 1 - ϵ, where ϵ << 1. (Hint: Use the binomial approximation.)

An antiproton p has the same rest energy as a proton. It is created inthe reaction p + p → p + p + p + p. In an experiment, protons at rest in the laboratory are bombarded with protons of kinetic energy Ek, which must be great enough so that kinetic energy equal to 2mc2 can be converted into the rest energy of the two particles. In the frame of the laboratory, the total kinetic energy cannot be converted into rest energy because of conservation of momentum. However, in the zero-momentumreference frame in which the two initial protons are moving toward each other with equal speed u, the total kinetic energy can be converted into rest energy. (a) Find the speed of each proton u such that the total kinetic energy in the zero-momentum frame is 2mc2. (b) Transform to the laboratory’s frame in which one proton is at rest, and find the speed u of the other proton. (c) Show that the kinetic energy of the moving proton in the laboratory’s frame is Ek = 6mc2 . Incidentally this tells you why…

Say that the mean lifetime of a π+ meson in its own rest frame is τ = 2.6 x 10-8 s. A pion of this lifetime is created at an altitude 100 km in the atmosphere by the collision of an incoming cosmic·ray proton with an atmospheric nucleus, and it has lifetime τ. Use the binomial approximation (see attached image) to determine how fast would this π+ meson need to move in order to reach the ground before decaying. Express the velocity in the form v/c = 1 - ϵ, where ϵ << 1.

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy 12.0×109 ly ly away is receding from us at 0. 0.900c , at what velocity relative to us must we send an exploratory probe to approach the other galaxy at 0.990c , as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy remains constant. (c) How long will it then take for a radio signal to be beamed back? (All of this is possible in principle, but not practical.)