In a colliding-beam experiment two beams of high-energy particles are made to cross each other. Since the collision occurs at relativistic speeds, the relevant measure of available energy is the center-of-mass frame.
For colliding beams, this is
( 
)
if the particles collide with equal and opposite momenta.
For a fixed target, it is
for a high-energy particle colliding with a target particle of mass "m."
The colliding-beam energy goes from
to
.
So, if you double the energy of the colliding beams,
you get double the collision energy.
On the other hand, the fixed-target energy goes from
to
.
Doubling the beam's energy in a fixed-target experiment only results in a
rise of available energy.
From this, it is obvious that
colliding beams are much more efficient than
fixed targets at getting high-energy collisions.
Question:
If you have two beams with energy = 50 GeV for a
colliding-beam collision, what energy is need for a
fixed-target collision with
a proton target
(
= 1 GeV) to get the same available energy?