Step 1 :Given masses and velocities: \(m = 1\), \(v = 1\), \(m_N = 2m\), \(v_N = v\), \(m_S = m\), \(v_S = 2v\), \(m_W = \frac{m}{2}\), \(v_W = 6v\), \(m_E = 3m\), and \(v_E = v\)
Step 2 :Calculate the momentum in the x-direction: \(p_x = m_Wv_W - m_Ev_E = \frac{1}{2}(6) - 3(1) = 0\)
Step 3 :Calculate the momentum in the y-direction: \(p_y = m_Nv_N - m_Sv_S = 2(1) - 1(2) = 0\)
Step 4 :Since both \(p_x\) and \(p_y\) are zero, the total momentum is zero.
Step 5 :\(\boxed{\text{D. The total momentum is zero.}}\)