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...
Here goes:
p = [E,ps] = energy-momentum four-vector for particle #1 in lab
frame.
[1,0] = energy-momentum four-vector for particle #2 in lab frame.
[4,0] = energy-momentum four-vector for RHS in CM frame.
E^2 - ps^2 = 1 (invariant mass of particle #1) [eq 4]
(E+1)^2 - ps^2 = mass of LHS = mass of RHS = 16
E^2 + 2E + 1 - ps^2 = 16 (by expanding the square)
2E + 1 = 15 (by subtracting eq 4)
E = 7
KE = 6
Note: no gammas involved. No Lorentz transformations.
Note: repeated, total reliance on the four-vector approach:
-- We add two four-vectors in the lab frame;
-- We add four four-vectors in the CM frame;
-- The norm of a four-vector is frame-independent.
(We use this idea at least three times.)