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Unformatted text preview: Chapter 7 Linear Momentum and Collisions 7.1 The Important Stuf 7.1.1 Linear Momentum The linear momentum of a particle with mass m moving with velocity v is deFned as p = m v (7.1) Linear momentum is a vector . When giving the linear momentum of a particle you must specify its magnitude and direction . We can see from the deFnition that its units must be kg m s . Oddly enough, this combination of SI units does not have a commonlyused named so we leave it as kg m s ! The momentum of a particle is related to the net force on that particle in a simple way; since the mass of a particle remains constant, if we take the time derivative of a particles momentum we Fnd d p dt = m d v dt = m a = F net so that F net = d p dt (7.2) 7.1.2 Impulse, Average Force When a particle moves freely then interacts with another system for a (brief) period and then moves freely again, it has a deFnite change in momentum; we deFne this change as the impulse I of the interaction forces: I = p f-p i = p Impulse is a vector and has the same units as momentum. When we integrate Eq. 7.2 we can show: I = i t f t i F dt = p 155 156 CHAPTER 7. LINEAR MOMENTUM AND COLLISIONS We can now defne the average force which acts on a particle during a time interval t . It is: F = p t = I t The value oF the average Force depends on the time interval chosen. 7.1.3 Conservation of Linear Momentum Linear momentum is a useFul quantity For cases where we have a Few particles (objects) which interact with each other but not with the rest oF the world. Such a system is called an isolated system . We oFten have reason to study systems where a Few particles interact with each other very briey, with Forces that are strong compared to the other Forces in the world that they may experience. In those situations, and For that brieF period oF time, we can treat the particles as iF they were isolated. We can show that when two particles interact only with each other (i.e. they are isolated) then their total momentum remains constant: p 1 i + p 2 i = p 1 f + p 2 f (7.3) or, in terms oF the masses and velocities, m 1 v 1 i + m 2 v 2 i = m 1 v 1 f + m 2 v 2 f (7.4) Or, abbreviating p 1 + p 2 = P (total momentum), this is: P i = P f . It is important to understand that Eq. 7.3 is a vector equation; it tells us that the total x component oF the momentum is conserved, and the total y component oF the momentum is conserved. 7.1.4 Collisions When we talk about a collision in physics (between two particles, say) we mean that two particles are moving Freely through space until they get close to one another; then, For a short period oF time they exert strong Forces on each other until they move apart and are again moving Freely. or such an event, the two particles have well-defned momenta p 1 i and p 2 i beFore the collision event and p 1 f and p 2 f aFterwards. But the sum oF the momenta beFore and aFter the collision is conserved, as written in Eq. 7.3....
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