Challenging problem involving special relativity

[Kyle.Nemeth]

Homework Statement

The creation and study of new and very massive elementary particles is an important part of contemporary physics. To create a particle of mass M requires an energy Mc². With enough energy, an exotic particle can be created by allowing a fast-moving proton to collide with a similar target particle. Consider a perfectly inelastic collision between two protons: an incident proton with mass m_p, kinetic energy K, and momentum magnitude p joins with an originally stationary target proton to form a single product particle of mass M. Not all the kinetic energy of the incoming proton is available to create the product particle because conservation of momentum requires that the system as a whole still must have some kinetic energy after the collision. Therefore, only a fraction of the energy of the incident particle is available to create a product particle. (a) Show that the energy available to create a product particle is given by Mc² = 2m_pc²√1 + K/2m_pc². This result shows that when the kinetic energy K of the incident proton is large compared with its rest energy mpc2, then M approaches (2mpK)1/2/c. Therefore, if the energy of the incoming proton is increased by a factor of 9, the mass you can create increases only by a factor of 3, not by a factor of 9 as would be expected.

Homework Equations

E² = (pc)² + (mc²)²

E = K + mc²

The Attempt at a Solution

I started this problem numerous times by modeling the two protons and the product particle as an isolated system and stating that p_i = p_f by the conservation of momentum. I then proceeded to use the energy-momentum relationship to find expressions for p_i and p_f. From there, I figured I'd be able to find the fraction of energy that goes into creating the product particle but after MANY many attempts, it seems I've just been running in circles. There seems to be something that I'm missing and I can't pinpoint what it is.

Here's what I did:

E_i² = p_i²c² + (2m_pc²)² is the initial energy of the proton-proton system.

E_f² = p_f²c² + (Mc²)² is the final energy of the proton-proton system.

I started by solving for p_i and p_f and then used p_i = p_f. From there I attempted to solve for Mc² but it seems that this has led me awry.

 

[my2cts]

Use units with c=1.
Given:
E1=sqrt(p^2+m^2)
p1=p
E2=m
p2=0
Use energy and momentum conservation to determine Ef and pf of a product particle with mass M:
Ef=sqrt(M^2+pf^2)