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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?

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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.

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