Question

The coefficient of restitution characterizes a collision in one dimension by relating the initial and final speeds of the particles involved,

$C_R = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$

In a 2D collision, the velocity can be split into components parallel and perpendicular to the "plane of collision" (the plane tangent to the two objects' surfaces at their contact point), and the equation above applies to the perpendicular components of the velocities only. One could write

$\begin{pmatrix}v_{2f\shortparallel} - v_{1f\shortparallel} \\ v_{2f\perp} - v_{1f\perp}\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & -C_R\end{pmatrix}\begin{pmatrix}v_{2i\shortparallel} - v_{1i\shortparallel} \\ v_{2i\perp} - v_{1i\perp}\end{pmatrix}$

My question is, is it useful (i.e. does it produce a more accurate description of realistic 2D collisions) to generalize that matrix? Perhaps by allowing the top left element to be unequal to 1, or allowing nonzero off-diagonal elements? Or is there some nonlinear relation that works better?

As usual, if you know of any relevant published research, references would be much appreciated.

Answer 1

To be simple, consider an object hitting a heavy plane. If there is coefficient of friction $C_F$ (and restitution $C_R$ ) between the bodies, parallel velocity is modified as well: $\Delta p_{\parallel} = F_{\parallel} \Delta t = -C_F F_{\perp} \Delta t = -C_F \Delta p_{\perp} = -C_F(1+C_R) p_{\perp}$

Thus obtaining

$\left[ \begin{array}{c} v_{\parallel}^{out} \\ v_{\perp}^{out} \end{array} \right] = \underbrace{ \left[ \begin{array}{cc} 1 & - C_F(1+C_R) \\ 0 & - C_R \end{array} \right]}_{A} \left[ \begin{array}{c} v_{\parallel}^{in} \\ v_{\perp}^{in} \end{array} \right].$

Note that it works as long as $v_{\parallel}^{in}\geq C_F(1+C_R) v_{\perp}^{in}$ . If there were a general (working for all input velocities) matrix A, it should allow dissipation, but not creation, of energy i.e. $||A||_2\leq 1$ (the condition is not fulfilled in the above example). I means that either the matrix need to have other entries as well or that the problem intrinsically needs restriction of the initial conditions.

However, bear in mind coefficient of restitution is only a effective parameter (as it has been already mentioned by jalexiou).

Anyway, it may be a nice experimental problem for an undergraduate student to find the all coefficients (and check for which conditions they work). I am curious of the results.