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Q.
A $1\, kg$ ball moving at $12\, ms ^{-1}$ collides with a $2\, kg$ ball moving in opposite direction at $24 \,ms ^{-1} .$ If the coefficient of restitution is $\frac{2}{3}$,
the their velocities after the collision are
$m _{1}=1 \,kg , \,\,u _{1}=12\, m s ^{-1}$
$m _{2}=3\, kg , \,\,u _{2}=-24\, m s ^{-1}$
$e =\frac{ v _{2}- v _{1}}{ u _{1}- u _{2}}$
$\frac{2}{3}=\frac{ v _{2}- v _{1}}{36}$
$v _{2}- v _{1}=\frac{36 \times 2}{3}$
$v _{2}- v _{1}=24 $ ... (1)
$m _{1} u _{1}+ m _{2} u _{2}= m _{1} v _{1}+ m _{2} v _{2}$
$1 \times 12+2 \times(-24)= v _{1}+2 v _{2} $
$12-48= v _{1}+2 v _{2}$
$v _{1}+2 v _{2}=-36 $
$\frac{- v _{1}+ v _{2}=24}{3 v _{2}=-12} $
$v _{2}=-4 \,m s ^{-1}$ [from equation (1)]
$-4- v _{1}=24$
$-4-24= v _{1} $
$v _{1}=-28 \,m s ^{-1}$