How is newtons third law of motion related to the law of momentum?

Answer 1

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.

Answer 2

Law of conservation of momentum states that the total momentum of system remain constant with time if there is no external force acts on a system of constant mass .Since this principle is conditional, its application needs careful attention to see that the system is free from any external exchange.This law is generalisation of third law of motion .

And , according to 3rd law of motion, to every action ,there is always opposed an equal reaction ,or, the mutual action of two bodies upon each other are always equal, and directed to contrary part.

Its application in recoil of gun:

Initially the gun and the bullet are at rest .Hence their initial total momentum of the system is zero .After firing the bullet of mass (m) moves with velocity (v) and the gun mass (m') kicks backward with a velocity (v') .Then from conservation of momentum

0 = mv + m'v' or v' = -mv/mv'.

negative signs means guns rebounds in backward direction which is 3rd law of motion.Also m<m' velocity of recoil is less than velocity of bullet.This is how Newton's Third law of motion related to conservation of momentum

Answer 3

Newton's 3rd law is the sum of the forces is zero, for every action there is an equal and opposite reaction. Fa + Fr = 0 = dPa/dt + dPr/dt = d(Pa + Pr)/dt = 0.

Thus the sum of the forces equal zero is the condition of the Conservation of Momentum.

Newton's Laws of Motion all derive from the Conservation of Energy. Conservation of Energy means Force is zero because the First Derivative of Energy is Force XE=F. Conservation of Energy is the First derivative is zero, thus force is zero.

Conservation of Momentum is the consequence of Conservation of Energy.