English: Proper-velocities (w ≡ dx/dτ = momentum/mass) are solid, while kinetic-energies (γ-1 ≡ dt/dτ-1 in units of mc²) are dashed. At low speeds (w<<c), collider velocities begin with a factor-of-two advantage over the accelerator value, while kinetic energies begin with a factor-of-four advantage. Also note that in both cases when w>>c, proper-velocities and kinetic-energies converge to the same dimensionless value (i.e. w/c ≈ γ-1).

Note that at low speeds, accelerator kinetic-energy (red, dashed) is approximately ½mw_AB² i.e has a slope of two on this log-log plot. This bends over to a slope of only one at high speeds as it approaches mw_ABc, while of course accelerator proper-velocity (red, solid) w_AB = w_AB has a slope of 1 at all speeds.

On the other hand, collider proper-velocity (blue, solid) has a slope of one (w_AC ≈ 2w_AB) which turns into a slope of two (w_AC ≈ 2w_AB²/c) at high speeds, while collider kinetic-energy (blue, dashed) has a slope of two for all speeds. Amazingly, in fact, collider kinetic-energy at any speed is simply (γ_AC-1)mc² = 4 × ½mw_AB².