Can the Bullet's Gamma Factor Exceed This Limit in Relativistic Collisions?

[MathematicalPhysicist]

consider a head on elastic collision of a bullet of rest mass M with a stationary target of rest mass m.
prove that the post-collision gamma factor of the bullet cannot exceed (m^2+M^2)/(2mM)

im given as a hint to prove that if P,P' are the pre and post 4 momentum of the bullet and Q,Q' are of the target, then in the CM frame (P'-Q)^2>=0
well i understand how from the hint i get the answer, but how to arrive at this, i mean the momentum is conserved, i.e: P'+Q'=P+Q
so by interchanging we get:
P'-Q=P-Q'
now in the cm, we must have: (P'-Q)^2-(P-Q')^2=(M+m)^2c^2
but i don't quite see why this is correct, i mean
P=(E/c,p)
Q=(mc,0)
P'=(E'/c,p')
Q'=(E''/c,p'')
where in the cm we have: p'=-p''
so (cause the 3 momentum there is zero) (P'-Q)^2=(E'/c)^2+(mc)^2-2*E'*m
(P-Q')^2=(E/c)^2+(E''/c)^2-2EE''/c^2
here's where I am stuck, can anyone help?

 

[Gokul43201]

Are you asking why the square of a real number must be non-negative? Maybe I'm misunderstanding...

 

[MathematicalPhysicist]

your'e correct, i thought about it myself, and there's not a lot to prove here, but then why is the hint to use the cm system to prove that (P'-Q)^2>=0?

 

[CompuChip]

Are you asking why the square of a real number must be non-negative? Maybe I'm misunderstanding...

P and P' are four-vectors, so I assume that (P - P')^2 is actually
[itex](P - P')_0^2 - (P - P')_1^2 - (P - P')_2^2 - (P - P')_3^2[/tex]
where the $P_i$ are real numbers, in which case it is not trivial at all that the "square" is positive.

 

[MathematicalPhysicist]

so how would you go around proving it?
my exam is tomorrow... (-:

 

[Gokul43201]

P and P' are four-vectors, so I assume that (P - P')^2 is actually
[itex](P - P')_0^2 - (P - P')_1^2 - (P - P')_2^2 - (P - P')_3^2[/tex]
where the $P_i$ are real numbers, in which case it is not trivial at all that the "square" is positive.

Next time, I'll actually read the question properly before I shoot my mouth... err, keyboard off!

 

[CompuChip]

I'm a bit confused by all the primes, I think you still need that
(E/c)^2+(E''/c)^2-2EE''/c^2 >= 0.
Can't you use some estimate like E > E'', which would give
(E/c)^2+(E''/c)^2-2EE''/c^2 >= (E/c)^2+(E/c)^2-2E^2/c^2 = 0
where the estimate is justified by a physical argument?