Elastic collision in one of the particles' frame

[Malamala]

Homework Statement

A particle B is standing still while another one, A, is moving towards it with initial 4-momentum $(E,p,0,0)$. Calculate the change in particle A's 4-momentum as viewed from the particle B's rest frame, in terms of the initial energy E and the scattering angle $\theta$.

Homework Equations

The Attempt at a Solution

I am a bit confused about the 4-momentum conservation. Initially we have $p^i_A=(E,p,0,0)$ and $p^i_B=(m_B,0,0,0)$ finally we should have $p^f_A=(E_f, p_f cos(\theta),p_f sin(\theta),0)$ and $p^f_B=(m_B,0,0,0)$. To get the change in momentum I would do $p^f_A-p^i_A$. But the total momentum should be conserved in any frame, but I am not sure how does that work here. In order to conserve it, we would need $E=E_f$ and $\theta=0$ but then the problem would be trivial and also physically you can obviously have angles other than 0. What am I doing wrong?

 

[mfb]

Particle B will move after the collision.

 

[maughanster]

mfb gave a good answer. After the collision, particle B is not stationary. Maybe this already clear, but the TOTAL momentum is conserved, not necessarily the momentum of each individual particle.