Answer
(a) $\frac{3}{7}v_{\circ}$
(b) $\frac{3}{7}$
Work Step by Step
(a) We can find the final speed of the carts as follows:
According to the law of conservation of momentum
$P_i=P_f$
$\implies mv_{\circ}+2mv_{\circ}+0=(m+2m+4m)v_f$
$\implies v_f=\frac{3}{7}v_{\circ}$
(b) We can find the required ratio of final and initial kinetic energy as follows:
$K_i=\frac{1}{2}(m+2m)v_{\circ}^2$
$K_i=\frac{3}{2}v_{\circ}^2$
and the final kinetic energy is given as
$K_f=\frac{1}{2}(4m+2m+m)(\frac{3}{7}v_{\circ})^2$
$K_f=\frac{1}{2}(7m)(\frac{3}{7}v_{\circ})^2$
Now
$\frac{K_f}{K_i}=\frac{\frac{1}{2}(7m)(\frac{3}{7}v_{\circ})^2}{\frac{3}{2}mv_{\circ}^2}$
$\frac{K_f}{K_i}=\frac{3}{7}$
