$$\require{cancel}$$
We shall often be interested in cases where we transfer both energy and momentum from one particle to another, i.e., we have inelastic collisions where particles change their character – e.g., their rest-mass. If we have, as in Figure $$\PageIndex{1}$$, two particles with energy-momentum $$k_1$$ and $$p_q$$ coming in, and two with $$k_2$$ and $$p_2$$ coming out, We know that since energy and momenta are conserved, that $$k_1+p_1=k_2+p_2$$, which can be rearranged to give $p_2=p_1+q,\quad k_2=k_1-q.$ and shows energy and momentum getting transferred. This picture will occur quite often!
Figure $$\PageIndex{1}$$: A sketch of a collision between two particles.