Given that

Let mass of nuclei at rest = M.

The masses of two smaller nuclei are m_1 \text{ and } m_2   

Velocities of the smaller nuclei are v_1 \text{ and } v_2

Prove that: Smaller nuclei must go in opposing directions.

Step 1: Calculating the momentum of the nuclei

Because the nuclei are at rest, their initial momentum is zero.

momentum of nuclei P = 0

Momentum of the smaller nuclei

P_1 = m_1v_1

P_2 = m_2v_2

Step 2: Prove that the velocities of smaller nuclei are in opposite direction

Use the conservation of the momentum

P = P_1 + P_2

0 = m_1v_1 + m_2v_2

m_1v_1 = - m_2v_2   

v_1 = - \frac{m_2v_2}{m_1}

v_1 \propto - v_2

Thus, we can conclude that both velocities are traveling in the opposite direction.