At a collider, two high-energy particles A and B with energies E_A and E_B, which are much greater than either of the masses, collide at a crossing angle θ. Show that the total centre-of-mass energy ECM is given by 

E²_CM = 2E_AE_B (1 + cos θ)

If the rest masses are neglected. At the HERA collider (which was operational until 2007) in the DESY laboratory in Hamburg, a 30 GeV electron beam collided with an 820 GeV proton beam at zero crossing angle. Evaluate the total centre-of-mass energy and show that a fixed-target electron accelerator would require a beam energy of approximately 5 × 10⁴ GeV to achieve the same total centre-of-mass energy.