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# l05 - lecture 5 Topics Where are we now Newton’s second...

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Unformatted text preview: lecture 5 Topics: Where are we now? Newton’s second law and momentum The third law Rocket motion Scattering and kinematics Elastic collisions Inelastic collisions The speed of a bullet More elastic collisions Where are we? In the previous lecture, we discussed conservation of energy. Today, after finishing what we didn’t get done on Thursday, I will talk about conservation of momentum and discuss the notion of scattering. I think this will be much easier, and we may spend much of the time in class on the material from last time. Newton’s second law and momentum Newton’s second law for a single particle of mass m can be written as ~ F = d~ p dt (1) where the quantity ~ p is the momentum of the particle, and is given in Newtonian mechanics by ~ p = m~v . (2) The form (1) actually turns out to be a better and more general way of writing the second law than the familiar ~ F = m~a . In this form, (1) [but not (2)] is true even when the speed of the particle approaches the speed of light, where as we will see in a few weeks, many of the common-sense aspects of mechanics begin to break down. In addition, as we will see, (1) allows us to deal more easily with situations in which objects come apart, or coalesce. The third law So if force is always changing momentum according to (1), how is it that momentum is conserved? The answer that you probably learned in high-school is Newton’s third law. For every action, there is an equal and opposite reaction. If thing 1 produces a force on thing 2, then thing 2 produces a force with equal magnitude and in the opposite direction on thing 1. If this is correct, then any change of the momentum of something is always compensated by a change in the momentum of the things that are producing the forces on it. The total momentum of any isolated system that has no external forces acting on it is always conserved. For now, you should just accept this. Later in the course (starting next week) we will talk more about why it is true. Here, we will be content to see how it is useful. 1 Rocket motion The most important uses of conservation of momentum all have a couple of things in common. The first is that using momentum conservation allows us to avoid thinking about incredibly complicated details of how the forces work that turn out to be irrelevant in the end. The second is that we make progress instead by comparing the system at two different times and using the fact that the momentum is the same. These two times may be far apart or close together. Sometimes we are interested in the initial condition of a system and the final condition, and we don’t much care about what happens in between. But sometimes, we are interested in comparing times that are very close together to use conservation of momentum to analyze the dynamics of the system. In the latter case, we can almost always analyze the problem by look at the difference between the system at time t and time t + dt . A good example of using momentum conservation to simplify the analysis....
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