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Newton's 3rd law requires two forces to be equal and opposite thus summing to zero.

The forces can be represented by the change in momentum thus requiring the momentum be conserved, dP/dt = 0.

0 = Fa + Fr =dPa/dt + dPr/dt =d(Pa + Pr)/dt =0.

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Newton's Laws are the manifestation of the Conservation of Energy. The 3rd Law is the vector part of the Conservation of Energy. Conservation of Energy requires that the sum of the Forces sums to zero.

If the forces sum to zero, and Force is the time derivative of Momentum, then the Momentum must be constant. Constant Momentum is the Conservation of Momentum.

Conservation of Momentum is a derivative of the Conservation of Energy and not an independent Law of Conservation as proposed by many including Emmy Noether.

The Conservation of Momentum is the vector part of the Quaternion Conservation of Energy. The Scalar part is said to be the Conservation of Energy. This confusion results from not recognizing that Physics is the science of Quaternion Quantities.

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