 Laboratory scattering circle

When a particle of mass $M_1$ and initial speed $u_1$ strikes a stationary mass $M_2$, the centre-of-momentum frame moves at $\mathbf{V}=M_1{u}_1\hat{\mathbf{x}}/(M_1+M_2)$. If the relative speed retention in an oblique collision is $q$, its possible outgoing velocity vectors lie on the displayed circle in a scattering plane. Intersect a ray of fixed laboratory direction with that circle to obtain the quadratic formula for outgoing speed. A negative root is inadmissible, and the origin has no defined scattering angle. Tangency gives the maximum laboratory angle when the circle excludes the origin.