A high speed proton collides with a proton at rest, two proton-antiproton pairs were created. That is some of the accelerated protons kinetic energy was enough to be converted into the mass of the new particles. If the proton-antiproton pairs move along together as a single particle of rest mass M=4m show that hte kinetic energy of the accelerated proton before collision is:
relativistic momentum: p=gamma(mv)
Relativistic kinetic energy: K=(gamma-1)mc2
energy as a function of momemtum: E2=(pc)2 + (mc2)2
The Attempt at a Solution
Heres my train of thought....So the kinetic energy of the moving proton must equal the mass energy of 2 additional particles in the product of the collision but those products ( the proton antiproton pairs) are also moving so thats additional kinetic energy. The reactancts have a mass of 2m and the products have a mass of 4m. The total momemtum must be the same since its conserved.
I tried figuring out the speed of the intital moving proton and the speed of the blob of 4 particles and got :
4u2 + v2=3c2
where u is the velocity of the proton and the v is the velocity of the 4 particle blob.
Then Iam not sure to procede from there. I am assuming that all the mass of the proton at rest was converted into energy but it doesnt really state that. It just seems wierd because the intital proton can moving at any speed greater than the least amount of speed required to create 4 particles that are moving...if that makes sense since it says that SOME of the accelerated protons kinetic energy was converted.
Thank you for your precious time.