Relativistic Electron/Positron collison.

[kirkbytheway]

Homework Statement

In an electron positron collider, an electron and a positron are collided such that their momenta
in the lab are equal and opposite. In such a collision, the electron and the positron annihilate
and produce a muon and an antimuon. The rest mass of a muon is 106MeV/c^2.
(a) What is the minimum electron energy in the lab necessary to produce the muon
antimuon pair, in MeV?
(b) Assuming that the electron rest energy can be neglected, what is the momentum
of the electron, in MeV/c ?
(c) What is the gamma factor for the frame of reference in which the electron is at
rest? Here you will need to use the electron rest mass, which is 0.51MeV/c^2.
(d) By performing a Lorentz transform, or otherwise, work out the energy of the
positron in the rest frame of the electron, in MeV. Assume that the positron mass can be
neglected. What relevance does this result have to the choice of colliding beam accelerators,
where the beams are bought together with equal energies and opposite momenta in the lab,
compared to fixed target accelerators, where a beam of particles is collided with target particles
at rest in the lab?

Homework Equations

2nd Year Undergraduate Special Relativity and non-calculator approximations.

The Attempt at a Solution

For part a) I got 106 MeV by converting the rest mass into a rest energy and working out the energy required for the muon-antimuon pair production which must be equal to the energy of the electron-positron collison and since the electron and positron have the same momenta then the electrons total energy is half of the collison energy.

For part b) I got 106 MeV/c by using the approximation E=pc since the electrons rest energy (and rest mass?) is neglible.

For part c) I got γ=208 from using γ=sqrt(1+(p/mc)^2).

Its part d) that I've hit a wall with. Since γ=208 then the particle is moving at over 99% of c. When I try to calculate the velocity of the positron in the electrons rest frame I get another value that is over 99%c. Is there a more algebraic approach that can prevent me from having to deal with long calculator displays. I'm also struggling with the qualatative part of the questions, am I right in thinking that these type of collisions are an easier way of reaching high-energies than the other types?

Thank you in advance for your help.

 

[clamtrox]

Its part d) that I've hit a wall with. Since γ=208 then the particle is moving at over 99% of c. When I try to calculate the velocity of the positron in the electrons rest frame I get another value that is over 99%c. Is there a more algebraic approach that can prevent me from having to deal with long calculator displays.

Yes, Taylor series are very useful if you want to avoid too many 9's. For example, you can use the velocity addition formula,
w = \frac{2v}{1+v^2/c^2}, then set your v almost equal to c, v = (1-x) c where x is small, substitute to equation above and expand as a series of x.