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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?
The problem is underspecified. Consider the collision in the centre of mass frame, where the particles have momentum $\vec{p}$ and $-\vec{p}$. Then so long as after the collision both particles come out in opposite directions with equal momenta both momentum and energy will be conserved. Thus you can't predict the final scattering angle from only the initial speeds and masses.
In the classical case where the objects are not truly pointlike but are hard spheres, the outcome is determined by the exact angle they initially collide at. This is pretty intuitive, with momenta transfer only possible along the normal to the plane of contact. Thus a perfectly head on collision results in a 1D collision, anything else leads to 2D scattering. In the quantum case there are some complications because things are genuinely pointlike and essentially you end up calculating a probability for each particle to scatter into any given angle.
1D is just a special case of 2D which is a special case of 3D. The difference is not physical but mathematical. Certain modes can be idealized as 1D while others cannot. Is the motion is defined along a line or on a plane or two skew lines?
