Momentum
Conservation

In a closed system (one that does not exchange any matter with its surroundings and is not acted on by external forces) the total momentum remains constant. This fact, known as the law of conservation of momentum, is implied by Newton's laws of motion. Suppose, for example, that two particles interact. As explained by the third law, the forces between them are equal in magnitude but opposite in direction. If the particles are numbered 1 and 2, the second law states that F = d p / d t and F = d p / d t. Therefore,

d p 1 d t = − d p 2 d t, {\displaystyle {\frac {{\text{d}}p_{1}}{{\text{d}}t}}=-{\frac {{\text{d}}p_{2}}{{\text{d}}t}},} with the negative sign indicating that the forces oppose. Equivalently,

d d t (p 1 + p 2) = 0. {\displaystyle {\frac {\text{d}}{{\text{d}}t}}\left(p_{1}+p_{2}\right)=0.}

If the velocities of the particles are v and v before the interaction, and afterwards they are v and v, then

m A v A 1 + m B v B 1 = m A v A 2 + m B v B 2. {\displaystyle m_{A}v_{A1}+m_{B}v_{B1}=m_{A}v_{A2}+m_{B}v_{B2}.}

This law holds no matter how complicated the force is between particles. Similarly, if there are several particles, the momentum exchanged between each pair of particles adds to zero, so the total change in momentum is zero. The conservation of the total momentum of a number of interacting particles can be expressed as m A v A + m B v B + m C v C +... = c o n s t a n t. {\displaystyle m_{A}v_{A}+m_{B}v_{B}+m_{C}v_{C}+...=constant.}

This conservation law applies to all interactions, including collisions (both elastic and inelastic) and separations caused by explosive forces. It can also be generalized to situations where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.
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