In my mechanics course, we've covered 1D and 2D collisions. However, what physical factors determine whether a collision will be 1 or 2 dimensional?
To make the question more precise, consider a mass $m_1$ with velocity $u_1$ approaching a stationary mass $m_2$. After the collision, $m_1$ and $m_2$ have velocities $v_1$ and $v_2$, respectively. What properties must $m_1$, $m_2$ and $u_1$ have for $u_1$, $v_1$ and $v_2$ to be parallel?