# Comparing the energies needed to reach the same relative velocity

I have a beam of particles being accelerated up to an energy $E$ and hitting a stationary target. I have 2 other beams pointed at each other and each one accelerates their respective particles up to $E_0<E$. How can I compare $E$ and $E_0$ in such a way as to show what values for each are needed to reach the same relative velocity?

You have to compute the energy in the (relativistic) centre of mass, generally indicated with $\sqrt s$. This is $2E$ (where $E$ is the beam energy) in the case of a high energy collider, but for fixed target it is just $\sqrt{2 E}$. The computation is pretty common in textbooks of accelerator physics and special relativity.

Here are some random online notes from a quick Google search.

The interesting fact is that although colliders allow keeping a linear dependency between beam energy and collision energy, they are much more complicated as they require extreme control of two beams. Also, the density of a beam is nothing compared to the one of a fixed target: most of the particles just go through the opposite beam without relevant interactions.