16-Special Relativity rev 4.0 . (Another Colliding Beam Accelerator) The Hera ac

Question

16-Special Relativity rev 4.0 . (Another Colliding Beam Accelerator) The Hera accelerator, which was decommissioned in summer 2007, consisted of a 30GeV (total) energy that collided head-on with an 820 GeV (total) energy proton beam electron beam a) P: Consider one electron colliding with one proton. What is the total energy and momentum of the system just before the collision? NOTE: This case is different than the LEP case in the previous problem as seern below) Suppose we make a new particle in these electron-proton collisions. Why can't this new particle be at rest? Can all of the systems initial energy be entirely converted into rest mass energy? Explain P: What is the maximum mass this new particle can have? Note: This is a simplified example. In real life, certain conservation laws (consérvation of baryon and lepton number) would not allow this process to occur unless a new fundamental force existed. You may ignore those laws in this problem. b) Qi c)

Explanation / Answer

13. given

eletron energy, Ee = 30 GeV

proton energy. Ep =820 GeV

a. one electron colliding with proton

hence total energy just before collision = E

E = Ep + Ee = 850 GeV

now, for momentum p, energy E

E^2 = p^2c^2 + mo^2c^4

hence

p^2 = (E^2/c^2 - mo^2c^2)

hence

total momentum P = Pe - Pp

P = Pe - Pp = sqrt(Ee^2/c^2 - me^2*c^2) - sqrt(Ep^2/c^2 - mp^2*c^2)

me = 9.1*10^-31 kg = 0.56875*10^-11 kg e

mp = 1.6*10^-27 kg = 1000*10^-11 kg e

hence

P = 99.9999999 - 2733.331686 eV/m/s

P = -4.213*10^-16 kg m/s

b. this new particle made in collision cant be at rest, as total momentum has to be conserved, which is non zero. hence the final signel particel has to be moving with some speed to keep this total moentum constant

hence the systems's entire energy cannot be converted into its rest mass energy as some energy will be needed as KE for the final particel to maintain its momentum as well

c. hence maximum mass of this particel = m

(850*10^9*1.6*10^-19)^2 = (4.213*10^-16 )^2*9*10^16 + m^2*c^4

m = 5.576*10^-25 kg

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