Conservation of Momentum

Conservation of Momentum - momentum of the particles P = p...

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Conservation of Momentum What happens when a group of particles are all interacting? Qualitatively speaking, each exerts equal and opposite impulses on the other, and though the individual momentum of any given particle might change, the total momentum of the system remains constant. This phenomenon of momentum constancy describes the conservation of linear momentum in a nutshell; in this section we shall prove the existence of the conservation of energy by using what we already know about momentum and systems of particles. Momentum in a System of Particles Just as we first defined kinetic energy for a single particle, and then examined the energy of a system, so shall we now turn to the linear momentum of a system of particles. Suppose we have a system of N particles, with masses m 1 , m 2 ,…, m n . Assuming no mass enters or leaves the system, we define the total momentum of the system as the vector sum of the individual
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Unformatted text preview: momentum of the particles: P = p 1 + p 2 + ... + p n = m 1 v 1 + m 2 v 2 + ... + m n v n Recall from our discussion of center of mass that: v cm = ( m 1 v 1 + m 2 v 2 + ... + m n v n ) where M is the total mass of the system. Comparing these two equations we see that: P = Mv cm Thus the total momentum of the system is simply the total mass times the velocity of the center of mass. We can also take a time derivative of the total momentum of the system: = M = Ma cm Recall also that, for a system of particles, F ext = Ma cm Clearly, then: F ext = Don't worry if the calculus here is complex. Though our definition of the momentum of a system of particles is important, the derivation of this equation only matters because it tells us a great deal about momentum. When we explore this equation further we will generate our principle of conservation of linear momentum....
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