# Conservation of momenta

1. Dec 11, 2007

### malawi_glenn

1. The problem statement, all variables and given/known data
An relativistic proton collides with a proton at rest (in Lab-frame), the collision is elastic.

let incoming proton have momenta p, and the outgoing momenta = p1, p2.

The following is conserved:

$$\vec{p} = \vec{p}_1 + \vec{p}_2$$

$$\sqrt{m^2+p^2} + m = \sqrt{m^2+p_1^2} + \sqrt{m^2+p_2^2}$$

Gives for the angle between p_1 and p_2 (in lab frame). A minima occurs, which means that $p1 = p2$. One can show that this minima occurs so that: $p1 + p2 > p$. Explain why that is possible!

3. The attempt at a solution

MATLAB

m = 0.93828; % proton mass in GeV

p = 2; %GeV incomming proton

p1 = [0.01:0.01:2]; %range of outgoin proton #1s momenta.

p2 = sqrt((sqrt(m^2+p^2)+m-sqrt(m^2+p1.^2)).^2-m^2);

omega = acos((p^2-p1.^2-p2.^2)./(2*p1.*p2));
omega = 180/pi*omega;

plotting gives minima för p = 1.2GeV/c

I am very unsure about this, I think it is possible $p1 + p2 > p$ scince momenta is a vector quantity, so the magnitudes can change, but not the total (i.e the total vector after = total vector initial). More suggestions?