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If a two body system has a zero total momentum, does it necessarily have zero kinetic energy?

No, a system can have zero momentum, but non-zero kinetic energy. Momentum is a vector quantity, and kinetic energy is a scalar quantity. Suppose a system contains two identical skate boards with riders of the same mass. If they are at rest relative to each other and facing each other, the system has zero momentum and zero kinetic energy. If they push against each other, they will go off in opposite directions at about the same speed. Since the momentum vectors are the same magnitude, but opposite direction, they add to zero. Their total momentum after the push is zero, just as it was before they pushed off each other. The momentum of the system is conserved. But each now has kinetic energy, a non-zero amount.

---Nathan A. Unterman

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