Question

Two balls of masses 2 kg and 4 kg are moved towards each other with velocities $4m/s$ and $2m/s$, respectively, on a frictionless surface. After colliding, the 2 kg ball returns back with velocity $2m/s$. Then find
a) velocity of the 4 kg ball after collision
b) coefficient of deformation $e$
c) impulse of deformation $J_d$
d) Maximum potential energy of deformation
e) impulse of reformation

Answer 1

a) By momentum conservation,
$2\left(4\right)-4\left(2\right)=2(-2)+4\left({\nu }_2\right)$
$⟹{\nu }_2=1m/s$.
b) $e=\frac{Velocityofseparation}{Velocityofapproach}=\frac{1-9-2)}{4-(-2)}=\frac{3}{6}=0.5$
c) At the maximum deformed state, by conservation of momentum, common velocity is $\nu =0$
$J_D=m_2(\nu -u_2)=4\left[\right(0-(-2)]=8Ns$
d) Potential energy at the maximum deformed state,
$U=$ loss in kinetic energy during defirmation
or $U=(\frac{1}{2}{m}_1{u}_1^2+\frac{1}{2}{m}_2{u}_2^2)-\frac{1}{2}(m_1+m_2){\nu }_2$
$=\left(\frac{1}{2}2\right(4{)}^2+\frac{1}{2}4\left(2{)}^2\right)-\frac{1}{2}(2+4)(0{)}^2=24J$
e) $J_R=m_2({\nu }_2-\nu )=4(1-0)=4Ns$
also $J_R=e{J}_D=0.5\times 8=4Ns$