Two perfectly elastic sphere of masses 2 kg and 3 kg moving in opposite directions with velocities after $8m{s}^{-1}$ and $6m{s}^{-1}$ respectively collide with each other. Find their velocities after the impact

Here $m_1=2kg,m_2=3kg,u_1=8m{s}^{-1},u_2=-6m{s}^{-1}$

According to the law of conservation of linear momentum; $m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

i.e., $\left(2\right)\left(8\right)+\left(3\right)\left(-6\right)=2v_1+3v_2\Rightarrow 2v_1+3v_2=-2\mathrm{...}\left(i\right)$

Since the collision is elastic, $u_1-u_2=v_2-v_1$

i.e., $8-\left(-6\right)=v_2-v_1\Rightarrow v_2-v_1=14\mathrm{.......}\left(iii\right)$

On solving the equations (i) and (ii) we get $v_1=-8.8m{s}^{-1},v_2=5.2m{s}^{-1}$