Question

A ban rolls without sliding over a rough horizontal floor with velocity $V_0=7m/s$ towards a smooth vertical wall. If coefficient of restitution between the wall and the ball is $e=07.$ Calculate velocity v of the ball long after the collision.

Answer 1

Between $A$ and $B$. there is forward slipping Therefore, friction
will be maximum and backwards (rightwards). At point B where $v=R\omega$ ball starts rolling without slipping and force of friction
becomes zero.
From conservation of angular momentum between points
$A$ and $B$ about bottommost point (because torque of friction about
this point is zero)
$L_A=L_B$
$\therefore m\left(0.7v_0\right)R-I{\omega }_0=mvR+I\omega$
Substituting ${\omega }_0=\frac{v_0}{R},\omega =\frac{v}{R}$
and $I=\frac{2}{5}m{R}^2,$ we get
$v=\frac{3}{14}{v}_0=\left(\frac{3}{14}\right)\left(7\right)m/s=1.5m/s$