Interpret the statements

• Statement $A$: Linear momentum of a system of particles is zero. $\vec P = \sum_i m_i \vec v_i = 0$

• Statement $B$: Kinetic energy of a system of particles is zero. $K = \sum_i \frac{1}{2} m_i v_i^2 = 0$

Check whether $A \Rightarrow B$

If total momentum is zero, does it mean total kinetic energy is zero?

Counterexample: Consider two particles of equal mass $m$ moving with equal speeds $v$ in opposite directions.

Their momenta are: $\vec p_1 = m\vec v, \qquad \vec p_2 = -m\vec v$ So total momentum is: $\vec P = \vec p_1 + \vec p_2 = 0$ But total kinetic energy is: $K = \frac{1}{2}mv^2 + \frac{1}{2}mv^2 = mv^2 \neq 0$

Therefore, $A \text{ does not imply } B$

Check whether $B \Rightarrow A$

If total kinetic energy is zero, does it mean total momentum is zero?

Since kinetic energy is a sum of non-negative terms, $K = \sum_i \frac{1}{2}m_i v_i^2$ can be zero only if each term is zero.

Hence for every particle, $v_i = 0$ So each particle’s momentum is also zero: $\vec p_i = m_i \vec v_i = 0$ Therefore total momentum is: $\vec P = \sum_i \vec p_i = 0$

Thus, $B \text{ implies } A$

Conclusion

• $A$ does not imply $B$
• $B$ does imply $A$

Hence the correct option is: $\boxed{\text{C}}$