Special Rel. Creates pions decaying, most of question completed.

[cpfoxhunt]

Homework Statement

I was wondering if I am getting anything like the right answers here to this:

12. A relativistic anti-proton, with total energy 30 GeV, travels 2.5 x i04 light years (2.36 x 1020 m) from the centre of the Milky Way galaxy and collides with a proton in the Earth's atmosphere How long does the journey take in the rest frame of the anti-proton? [3]

How much energy is available for the production of new particles from this collision? [8]

Assuming that all of this available energy is used in the production of pions in a single event, how many pions would be generated? The pions travel vertically downwards through the Earth's atmosphere. A counter in a laboratory on the Earth's
surface records 7 pions from the collision event. Estimate the height above the laboratory at which the collision occurred. [9]

[The rest mass of a pion is 135MeV/c2, and that of a proton is 938MeV/c2. The mean
proper lifetime of the pion is 2.5 x 10-6 s.

Homework Equations

Invarience of the interval, e^2 - p^2 = m^2

The Attempt at a Solution

OK, first I use the fact that E = MGamma to work out that gamma is about 31.98, and beta is about 0.99 so the thing is traveling at near as can tell c ( well 0.999c, nothing that makes any difference). In the rest frame of te anti proton it won't take nearly so long because the galazy is moving pretty quickly past it, so t = 2.5E4/31.98 = 738 years

Now for the energy: I have a particle with the energy of the rest mass of the proton (I assume the one in the Earth isn't doing much), and a particle with 30Gev. M = mass proton, assume particle physics units. E1 = 30Gev

Using invarience, (E1 + M)^2 -P1^2 = Etotal squared

Using the fact that E1^2 - p1^2 = M^2

2E1M + 2M^2 = E total squared

So E total available = 7.6 Gev

This can make abot 56.4 pions (so call it 56 pions)

This is where I start having problems: For that 56 pions to decay to 7 pions, i can work out how much time has passed (mean lifetime * ln2 = half life, 3 half lives have passed so about 5.19E-3 seconds).

How do I work out how fast the pions are going to finish the question? If I assume they are going at c I get a fairly sensible answer, but why would I be able to assume that?

Cheers
Cpfoxhunt

 

[ijustlost]

Remember that this time (5.19E-3) is the time that elapses in the pions rest frame, not the Earth's rest frame. In the Earths rest frame elapsed time will be longer.
I don't think you can just assume the pions move at c.
What you do know is the Total Kinetic of all the pions T = (Original Energy - 7.6 Gev). Since we're assuming all available energy went to producing the pions, the pions must all move at the same velocity. So each pion has the same KE (T / #pions). Then you can find their velocity