4.2.1. colliders vs. accelerators

Proper-velocity w ≡ dx/dτ = γv can be written as Lorentz-factor γ ≡ dt/dτ times coordinate-velocity v ≡ dx/dt. For unidirectional motion, proper-velocities simply add by the symmetric relation: ,

i.e. the coordinate-velocities add while the Lorentz-factors multiply. 

Reporting accelerator speeds in proper units (e.g. of 1 ly/ty) of course makes these more comparable (e.g. in terms of momentum, energy, etc.) to other projectile speeds, but also illustrates the collider advantage much more graphically. Colliders (some with Lorentz-factors well over 105) may thus explore much higher-speed collisions than would be possible with a fixed-target accelerator. 

Example Problem 4.2.1a: What's the proper land-speed-record for protons accelerated in the laboratory on earth? How about for electrons, or for iron nuclei?

Example Problem 4.2.1b: If two of the protons mentioned in the previous problems were put onto a head-on collisions course, what would be the relative proper-speed of collision, obtained at likely less than twice the cost of running one of those protons into a stationary target?