Calculating Centre of Mass Energy in Fixed Target Interactions with the LHC

[leoflindall]

Homework Statement

The proposed maximum colliding beam energy, E_c, in the Large Hadron Collider is 7TeV per beam (proton-proton collisions). By a Lorenz transformation, show clearly that to produce the same centre of mass energy in a 'fixed target' interaction, the beam energy woulr have to be $$\frac{2E^{2}_{c}}{M_{p}}$$, where m_p is the proton mass and it is assumed that the beam energy is very much greater then M_pC².

Homework Equations

E=mc²

E=pc

E²=p²c² + (mc²)²

The Attempt at a Solution

I don't think this is particularly hard but I can't see how to approach this problem. I though maybe take the total relativistic energy to be equal to the kinetic energy + the rest energy, and then rearrange for the required equation but I can't get it to work for me.

I would appreciate if anyone could tell em how to approach this problem?

Many Thanks

Leo

 

[henry_m]

The key thing to remember is that for any system with total energy E and momentum p, the quantity E²-c²p² is a scalar (frame independent): the squared rest energy.

You need to work in two frames: the centre of mass frame, and the frame in which one beam is at rest. Equate the invariant rest energy for each single beam and for the whole system in each frame.

 

[leoflindall]

Cheers for the help! Worked it out now!

Many thanks

Leo