Answer
(a) $\frac{v_{\circ}}{2}$
(b) $\frac{v_{\circ}}{\sqrt 2}$
Work Step by Step
(a) We know that
$P_i=mv_{\circ}+2m(0)=mv_{\circ}$
and $P_f=m(0)+2mv_f=2mv_f$
Now according to the law of conservation of momentum
$P_i=P_f$
$\implies mv_{\circ}=2mv_f$
This simplifies to:
$v_f=\frac{v_{\circ}}{2}$
(b) We know that
$K.E_i=\frac{1}{2}mv_{\circ}^2+\frac{1}{2}(2m)(0)^2=\frac{1}{2}mv_{\circ}^2$
and $K.E_f=0+\frac{1}{2}(2m)v_f^2=\frac{1}{2}(2m)v_f^2$
Now according to the law of conservation of energy
$K.E_i=K.E_f$
$\implies \frac{1}{2}mv_{\circ}^2=\frac{1}{2}(2m)v_f^2$
This simplifies to:
$v_f=\frac{v_{\circ}}{\sqrt 2}$
