The balls, having linear momenta $\overrightarrow{\mathrm{p}}_1=\mathrm{p} \hat{\mathrm{i}}$ and $\overrightarrow{\mathrm{p}}_2=-\mathrm{p} \hat{\mathrm{i}}$, undergo a collision in free space. There is no external force acting on the balls. Let $\vec{p}_1$ and $\vec{p}_2$ be their final momenta. The following option(s) is (are) NOT ALLOWED for any non-zero value of $\mathrm{p}, \mathrm{a}_1, \mathrm{a}_2, \mathrm{~b}_1, \mathrm{~b}_2, \mathrm{c}_1$ and $\mathrm{c}_2$.