A 1kg ball moving at $12{\text{ms}}^{-1}$ collides head on with a 2kg ball moving in the opposite direction at $24\text{m/s}$ . The velocity of each ball after the impact, if the coefficient of restitution is 2/3, is

${\text{m}}_2=2\text{kg,}{\text{u}}_2=-24\text{m/s}$

$\text{e}=\frac{2}{3}$

${\text{v}}_{\text{1}}=\frac{\left({\text{m}}_{\text{1}}-{\text{em}}_{\text{2}}\right){\text{u}}_{\text{1}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}+\frac{\left({\text{m}}_{\text{1}}\right)\left(1+\text{e}\right)\left({\text{u}}_{\text{2}}\right)}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}$

$=\frac{\left(1-\frac{4}{3}\right)12}{3}+\frac{1\left(1+\frac{2}{3}\right)\left(-24\right)}{3}$

$=\frac{\frac{-1}{3}\times 12}{3}+\frac{2\times \frac{5}{3}\times -24}{3}$

$=\frac{-4}{3}+\frac{-80}{3}$

$=\frac{-84}{3}$

${\text{v}}_{\text{1}}=-28\text{m/s}$

${\text{v}}_2=\frac{{\text{m}}_{\text{1}}\left(1+\text{e}\right){\text{u}}_{\text{1}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}+\frac{\left({\text{m}}_{\text{2}}-{\text{em}}_{\text{1}}\right){\text{u}}_{\text{2}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}$

$=\frac{1\left(1+\frac{2}{3}\right)12}{3}+\frac{\left(2-\frac{2}{3}\right)}{3}-24$

$=1\times \frac{5}{3}\times 12\times \frac{1}{3}+\frac{4}{3}\times -24\times \frac{1}{3}$

$=\frac{20}{3}+\left(\frac{-32}{3}\right)$

${\text{v}}_{\text{2}}=\frac{-12}{3}=-4\text{m/s}$