Mandelstam variables consider the relativistic two-body scattering a+b→c+d with corresponding 4-vectors pa, p pc, and pp the masses of the corresponding objects are m,', mb, me, and mb. you can set the speed of light as c l if you would like. the mandelstam variables are useful lorentz invariants: t = (pa-pc) u = (pa-po) (a) show that s + t + u = mhng + mる+ m2, . [ 10 points] (b) show that the total center of mass energy is given by s. [10 points] (c) the lhc collides protons in the lab frame with an energy of 6.5 tev against other protons with an energy of 6.5 tev (so that 's = 13 tev) one of the things looked for is the production of new z bosons. what is the most massive z' that the lhc could directly observe? [10 points] (d) why can the photon not decay? [10 points] hint 1: prove that the mass of a decaying particle must exceed the masses of the particles produced in its decay. in other words, if a decays into b+c, show that m,> mmc. hint 2: evaluate in the center of mass frame.

Which types of electromagnetic waves have higher frequencies than the waves that make up ultraviolet light? check all that apply. radio waves infrared light microwaves gamma rays visible light x-rays

Some material consisting of a collection of microscopic objects is kept at a high temperature. a photon detector capable of detecting photon energies from infrared through ultraviolet observes photons emitted with energies of 0.3 ev, 0.5 ev, 0.8 ev, 2.0ev, 2.5ev, and 2.8ev. these are the only photon energies observed. (a) draw and label a possible energy-level diagram for one of the microscopic objects, which has four bound states. on the diagram, indicate the transitions corresponding to the emitted photons. explain briefly. (b) would a spring–mass model be a good model for these microscopic objects? why or why not? (c) the material is now cooled down to a very low temperature, and the photon detector stops detecting photon emissions. next, a beam of light with a continuous range of energies from infrared through ultraviolet shines on the material, and the photon detector observes the beam of light after it passes through the material. what photon energies in this beam of light are observed to be significantly reduced in intensity (“dark absorption lines”)? explain briefly.

This is important 1. what happens to the speed of molecules in water vapor when it condenses to form a liquid? (2 points) the molecules slow down. the molecules speed up. the molecules slow down and then speed up. the molecules speed up and then slow down. 2. how might human activities change to compensate for a decrease in water supplies? (2 points) humans might build a factory. humans might build drainage systems. humans might extract more water from the ground. human behavior does not change because of the water cycle. i'll be grateful if anyone can answer these 2 questions !