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Physics: Linear Momentum and Impulse

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About this Lesson

  • Type: Video Tutorial
  • Length: 10:56
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 118 MB
  • Posted: 07/02/2009

This lesson is part of the following series:

Physics (147 lessons, $198.00)
Physics: Momentum (8 lessons, $16.83)
Physics: Momentum and Its Conservation (5 lessons, $9.90)

This lesson was selected from a broader, comprehensive course, Physics I. This course and others are available from Thinkwell, Inc. The full course can be found at http://www.thinkwell.com/student/product/physics. The full course covers kinematics, dynamics, energy, momentum, the physics of extended objects, gravity, fluids, relativity, oscillatory motion, waves, and more. The course features two renowned professors: Steven Pollock, an associate professor of Physics at he University of Colorado at Boulder and Ephraim Fischbach, a professor of physics at Purdue University.

Steven Pollock earned a Bachelor of Science in physics from the Massachusetts Institute of Technology and a Ph.D. from Stanford University. Prof. Pollock wears two research hats: he studies theoretical nuclear physics, and does physics education research. Currently, his research activities focus on questions of replication and sustainability of reformed teaching techniques in (very) large introductory courses. He received an Alfred P. Sloan Research Fellowship in 1994 and a Boulder Faculty Assembly (CU campus-wide) Teaching Excellence Award in 1998. He is the author of two Teaching Company video courses: “Particle Physics for Non-Physicists: a Tour of the Microcosmos” and “The Great Ideas of Classical Physics”. Prof. Pollock regularly gives public presentations in which he brings physics alive at conferences, seminars, colloquia, and for community audiences.

Ephraim Fischbach earned a B.A. in physics from Columbia University and a Ph.D. from the University of Pennsylvania. In Thinkwell Physics I, he delivers the "Physics in Action" video lectures and demonstrates numerous laboratory techniques and real-world applications. As part of his mission to encourage an interest in physics wherever he goes, Prof. Fischbach coordinates Physics on the Road, an Outreach/Funfest program. He is the author or coauthor of more than 180 publications including a recent book, “The Search for Non-Newtonian Gravity”, and was made a Fellow of the American Physical Society in 2001. He also serves as a referee for a number of journals including “Physical Review” and “Physical Review Letters”.

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In physics, we often define quantities, like velocity was defined to be X over T, or kinetic energy, it's mv squared. We make these definitions not just to randomly combine symbols, but because they have physical significance. Kinetic energy is a quantity that teaches us something about how systems behave. It's got physical meaning.
So I'm going to define a new combination of symbols. Right now, it's just a definition: momentum. It's got a name. And it's given the symbol p and it's defined to be mass times velocity. So you take an object and you can define its momentum. I don't know why we use the symbol p. I just think we didn't want to call it m, because we're already using the symbol m. It's a useful quantity, but it's not obvious when you look at the equation. Why would this be useful? If you stare at it and think about it a little bit, you can sort of get a physical sense for what momentum is. If you take an object and you toss it around, this thing has some oomph to it. It's traveling and it hits my hand. Now, what do I mean by oomph? It's a feeling that I've got. Is it oomph of velocity? Certainly, when it's going faster, it's got more oomph, but it's not just velocity, because if I take a little light object and toss it around, even at high velocity it doesn't have as much oomph as the heavy one. So mass times velocity certainly seems to carry some physical insights with it.
We can see another hint about why momentum is a useful physical quantity by thinking about Newton's Second Law, the center pin of dynamics and understanding how things move and why. F equals ma, force equals mass times acceleration, and acceleration is defined to be a derivative of velocity, with respect to time, dv dt, change in velocity with time. This is Newton's Law the way we've been writing it. The way Newton himself wrote it was sliding the m inside the derivative. As long as you have an object with a definite mass, you're free to do so. And what you get then is derivative of mv with time. It's the same equation and mv, there it is, momentum. Force is dp dt. That is, in fact, the way Isaac Newton though about the Second Law and it's correct. We will discover that when mass is changing, like with a rocket that's ejecting mass, this is the form of Newton's Second Law that you really want to use. So we've actually gotten to a more fundamental law of physics by introducing this idea of momentum. And momentum is a property of an object. If you know the velocity and the mass, you know the momentum. And then it's the change in that property with time that tells you about forces, and vice versa. If you know the force, it teaches you immediately about the change in momentum. So you get some sense that momentum might be physically meaningful and we're going to be seeing, as we go into future tutorials, just exactly what circumstances momentum really becomes physically important and useful for problem solving.
For right now, let's just look at some more properties related to momentum. The change in momentum with time is force, so change in momentum, p, is something that we really might care about in a physical situation. You start off with a certain mv, and then somebody does something, maybe you crash into something. You have a collision and there's a change in momentum. And we might like to know what that is. Well, since F is p over t, this is the average expression when I use deltas instead of derivatives; this is the average force over some time interval. You can see that the change in momentum is just the average force times the length of time. So a force acting over a long time period will make a big change in momentum.
Change is momentum is clearly also physically interesting. It's what you care about in a lot of problems. It's interesting enough that people have given it a name and it's called I for impulse. The impulse is the change in momentum of an object, it's the average force times time. You don't have to work with average forces; we have the derivative formula. If force is changing all the time, rather than using this expression, I could just write the integral of Fdt. So let me just clean things up here. Impulse is a vector, it's the integral of force times time from t[initial] to t[final], and it's the change in momentum from t[initial] to t[final]. So impulse is certainly physically interesting. If you've got an object and it starts off in some momentum state and it ends up in another one, you'd like to be able to predict what the change in momentum is.
Let's think about a situation where impulse might be a useful quantity. Imagine that you're golfing and you're whacking the golf ball, so there's a collision between the golf ball and the club. And what are you interested in? Well, there's various things you might be interested in, but the most important thing, probably as a golfer, is the initial momentum was zero, because velocity was zero, and then it goes off flying and you'd like to know how fast and in what direction. You would like to know the change in momentum. So you want to know this integral, integral of Fdt. Now, let's think about what a graph of force versus time would look like in a collision of golf club with golf ball.
At some time t[i] you make contact. The force rapidly grows, and then it rapidly gets smaller again as the golf ball leaves the club, and then it's over. The collision is done and the ball is flying and it's got some momentum. And, from now on, it's just simple Newton's Law. If you were out in space, it would just go in a straight line forever. In the earth's gravitational field, it'll follow a simple parabolic trajectory that we know about. This is the interesting part of the story, and it's complicated. It's a short amount of time, the force is changing wildly with time and it's obviously going to be really tough to measure the force as a function of time. You need some little device that doesn't break when you hit it. What do we care about? We care about the impulse. Impulse is the integral of force times time. It's the shaded area under this curve that's physically meaningful. And so you could look at this curve and try to analyze all the complicated details. It might be easier to think about the average force acting over that time interval. Average force, remember, is just p over t. It's the force that, if it were constant and acted over this time period, you would get the same area, the same impulse. So the area under this gray rectangle and the area under the complicated purple curve are the same, it's the same impulse. So, to the outside world watching, that's what we care about.
We're going to work out techniques to calculate impulse and use it in problems. Let me give you a physical demonstration of the significance of this connection between impulse and momentum and force. Take an egg and drop it. If I'm careful, I can catch it. What do I do to catch it? I let my hand collapse, just the slightest little bit. Let me drop it into this bowl, same initial height. You know what's going to happen. The egg smashes. Something different, something significantly different has happened. I'm glad this didn't happen in my hand. What's the difference between these two situations? In both cases, the object was freely falling, so the moment before I caught it, in either case, its velocity was the same. You could determine that with simple one-dimensional kinematic. So, in both cases, it had a downward momentum. And then I caught it or stopped it. So, in both cases, change in momentum was exactly the same, t[final] minus p[initial]. So what was different in the two cases? It's not the change in momentum or the impulse that was different in those two cases. Remember force is p over t. The average force is change in momentum divided by the amount of time it took. With you hand, this little brief collapse of your hand increases the amount of time over which the momentum changes. When it hits the glass bottom, the amount of time is very short. Same p, but very different t means very different forces. When t is short, the average force is big and the egg breaks. So it's this connection between momentum and then change in momentum, which is impulse, and then the rate of change of momentum, which is force, that really is the physics that we're after.
I've defined these two new quantities, momentum and impulse, which is change in momentum. We've got some formulas for them. When are we really going to use them? I'm not really going to make use of this new concept in the same old problems that we've been working before, where you have a single object acting with some external force or forces. That's not usually a case where you need this new stuff. We could already take care of such problems with Newton's Law and conservation of energy and so on. This really comes into its strength, this new idea, when we talk about collisions, where you have rapidly changing forces over short periods of time. And it's difficult to think about forces. It's difficult to apply Newton's Law the way we have been. It's going to be easier to think about momentum and change in momentum. And we'll be able to solve lots of problems, collision problems, and also problems when you have systems of particles, like the golf ball and the golf club together, two particles, and you want to think about both of them together. We'll see that momentum is really the right idea to use in such circumstances.
Momentum
Momentum and Its Conservation
Linear Momentum and Impulse Page [1 of 2]

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