Answer
We can rank the groups according to center-of-mass speed, from the greatest to the least:
$(d) \gt (c) \gt (a) \gt (b)$
Work Step by Step
We can find an expression for the center of mass speed in each case.
(a) Along the x axis:
$P_x = \sum m_i~v_i = M~v_{com,x}$
$v_{com,x} = \frac{\sum m_i~v_i}{M}$
$v_{com,x} = \frac{mv-mv+0}{3m}$
$v_{com,x} = 0$
Along the y axis:
$P_y = \sum m_i~v_i = M~v_{com,y}$
$v_{com,y} = \frac{\sum m_i~v_i}{M}$
$v_{com,y} = \frac{-mv+0+0}{3m}$
$v_{com,y} = -\frac{v}{3}$
We can write an expression for the speed of the center of mass:
$v_{com} = \frac{v}{3}$
(b) Along the x axis:
$P_x = \sum m_i~v_i = M~v_{com,x}$
$v_{com,x} = \frac{\sum m_i~v_i}{M}$
$v_{com,x} = \frac{mv-mv+0+0}{4m}$
$v_{com,x} = 0$
Along the y axis:
$P_y = \sum m_i~v_i = M~v_{com,y}$
$v_{com,y} = \frac{\sum m_i~v_i}{M}$
$v_{com,y} = \frac{mv-mv+0+0}{4m}$
$v_{com,y} = 0$
We can write an expression for the speed of the center of mass:
$v_{com} = 0$
(c) Along the x axis:
$P_x = \sum m_i~v_i = M~v_{com,x}$
$v_{com,x} = \frac{\sum m_i~v_i}{M}$
$v_{com,x} = \frac{mv-mv+0+0}{4m}$
$v_{com,x} = 0$
Along the y axis:
$P_y = \sum m_i~v_i = M~v_{com,y}$
$v_{com,y} = \frac{\sum m_i~v_i}{M}$
$v_{com,y} = \frac{-mv-mv+0+0}{4m}$
$v_{com,y} = -\frac{v}{2}$
We can write an expression for the speed of the center of mass:
$v_{com} = \frac{v}{2}$
(d) Along the x axis:
$P_x = \sum m_i~v_i = M~v_{com,x}$
$v_{com,x} = \frac{\sum m_i~v_i}{M}$
$v_{com,x} = \frac{mv+mv+0+0}{4m}$
$v_{com,x} = \frac{v}{2}$
Along the y axis:
$P_y = \sum m_i~v_i = M~v_{com,y}$
$v_{com,y} = \frac{\sum m_i~v_i}{M}$
$v_{com,y} = \frac{-mv-mv+0+0}{4m}$
$v_{com,y} = -\frac{v}{2}$
We can write an expression for the speed of the center of mass:
$v_{com} = \sqrt{(\frac{v}{2})^2+(-\frac{v}{2})^2} = \frac{\sqrt{2}~v}{2}$
We can rank the groups according to center-of-mass speed, from the greatest to the least:
$(d) \gt (c) \gt (a) \gt (b)$
