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Physics Problems: Conservation of Momentum


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About this Lesson

  • Type: Video Tutorial
  • Length: 12:40
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 137 MB
  • Posted: 07/01/2009

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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Whenever you have a system of particles and the net external force acting on it is zero, then the total momentum of your system is going to be conserved. Mathematically, no net external force means total momentum is conserved or, if you want, you can write that there's no change in the total momentum. Conservation of a quantity means it's always the same, there's no change. And another way to say it, the initial momentum and the final momentum will be equal, whatever you call initial, whatever you call final. These are vector equations and they all say the same thing. Remember total momentum of a system means you have to add-up the momentum of all the little bits and pieces, add them up like vectors, and that grand total will be conserved. It's nice when you have a conserved quantity, because you figure out what it is in the beginning of the problem. All sorts of complicated stuff happens, and then, no matter what, at the end of the story the total quantity, in this case momentum, is still going to be the same as it was in the beginning.
Let's look at an example. You can think of this as two train cars on a train track, two air carts on an air track. There's no friction here, so we don't have to worry about external forces, and I'm going to send this one coming in, moving to your right, and there's going to be a collision, complicated collision. These are springs any more, it's a little ball of wax, so they're going to stick together. And what's going to happen? Let's just watch. This one's at rest, this one comes flying in, stick together. What you should notice is that it came moving in quickly, stuck together, and then they moved together slowly. Can we figure out exactly what the final velocity was? That's a typical physics goal; you know what was happening in the beginning, can we predict what happens in the end? So you think, "How could I do that?" Newton's Law, that's the sort of first guess, Newton's Second Law, that's how you always solve problems in physics. So you think, "Well, I'm in trouble here, because I don't know anything about the details of the force, complicated little force here. I don't really know what the formula for it is, so I don't think I, practically speaking, can use F = ma."
How about conservation of energy, initial energy equals final energy? No, that doesn't work here either. And this is a little bit subtler. When you've got whacks and things sticking together, there's friction. And whenever there is friction, there is going to be a loss of energy. And it may or may not be obvious, but it's a fairly large, substantial loss of energy here. You can't use conservation of energy in problems when friction is an important part of the story. Its frictional forces dissipate kinetic energy into thermal energy.
So what am I left with? No external forces, we can use conservation of momentum. One thing you might worry about, there is an external force; there's gravity, but there's also a normal force of the track, and those two add up to zero, of course, because it's not rising up or sinking down. So the net external sum of those two forces is zero, so we're gold. Momentum is conserved. Let's write it down mathematically and derive for the final condition, given the initial. So here's a picture. Math number one is moving to the right, with initial velocity v striking an object at rest, m2. And here's the final situation, m1 and m2 are hooked together and traveling off with final velocity vf. So let's write down conservation of momentum. That's a vector equation. I want to write down the x component of that vector equation, so let me pick a coordinate system. I'll call right positive, so the x component of momentum, Pi in the x direction, is m1v1 plus m2 times zero. This one's at rest. That's the initial momentum. Pf in the x direction is - here's the picture, m1 and m2 are both moving to the right with vf, so I could write this as m1vf plus m2vf or, if you prefer, you could just think of it a system with mass m1 plus m2 moving to the right with velocity vf. Initial momentum, final momentum, they're equal to one another by conservation of momentum. So I solve for vf and I get the following very simple result: vf is equal to m1 divided by m1 plus m2 times vi.
So this is a very specific formula to this particular problem. It tells me an awful lot. First of all, the sign is the same on both sides. So, no matter what, if this particle is moving to the right in the beginning, the whole system is going to be moving to the right at the end. That's conservation of momentum. In this case, the m's were equal. If m1 equals to m2, then I've got m1 over 2m1, that's , so vf is of the vi. And that's what we saw. We came in with some vi, they stuck together and they went out with half of the initial speed.
Let's look at some other cases. Supposing that m2 was very large. So let me add a whole bunch of weight to m2. So I've got a big old denominator, same small numerator, so this becomes a small positive number. m1 over m1 plus m2 is always less than or equal to one, but now it's really small. So I predict from this formula that vf should be very slow, but still to the right. Make sure there's no friction, send it in with a high speed, they couple together and, sure enough, they're moving off in this direction slowly. You may have noticed they really appear to be slowing down. Probably during that collision, they bent a little bit and there was a little external friction. If you can keep external friction out of the story, this equation is not approximate, it's exact. This is derived from Newton's Laws. It's a fundamental principle of nature that momentum is conserved if there's no external force.
Let's look at another limit here. What happens if I put the big mass on m1? So now m1 is the big one. What happens here? I've got a big number plus a small one, so big plus small, m2 becomes negligible. It looks like m1 over m1. It basically looks like vf is approximately the initial, only a little bit less. So, let's see if that's true. This one's at rest, just cruises right along. As the formula says, vf was very close to vi, a little bit of slowdown. According to the formula, that's it.
You can use conservation of momentum in lots of problems. It doesn't have to be collision problems. Here's another classic example. Supposing you launch a firework. So right at the very top, it blows up. Now, let's think of a simple firework that just splits into two. Depending on how much explosive is in there, just about anything could happen. You could have a dud, and so if they were heading initially with vi, they kid of split apart and they're still continuing, both pieces with vi. Let me show you a picture here. Here's the firework, initially two pieces of mass m heading to the right with vi, and then boom, it blows up and you've got two chunks m and m. And they have some final velocities. What I'm saying is, depending on the amount of explosives, anything could happen. Conservation of momentum alone is not enough to predict all the details of what's going to happen. If the explosion was huge, the back piece may go flying backwards and the front piece may go flying forward. However, I can pick the explosion to be just so, that the back piece just comes to a halt and the front piece is flying off. So I have to add one more bit of information before we can solve the problem, and that bit of information is I'm telling you that I've chosen the explosion so that the back end has come to a halt. So what's vf in this situation going to be? How fast will the front end go? It seems like that one bit of information isn't enough, I still haven't told you about the forces, I still haven't told you about the energy - conservation of momentum. We had an initial momentum, we've got a final system with the same momentum. Let's write down initial momentum in the x direction. It's mvi plus mvi, it's 2mvi or vinitial. And how about afterwards, Pf in the x direction? Remember this piece is at rest and this piece is moving with vf. (Pf)x is mvf and we set them equal to one another and I see the m's cancel and vf is twice the vi. So indeed, conservation of momentum is all I needed to solve this problem without having to worry about all these potentially scary details. You might have thought, "There's no way we could solve this problem."
I use conservation of momentum, it's worth just thinking to ourselves, "Was it valid? Was there really no net external force in the problem?" It's a firework and there is gravity. Gravity is an external force. Why didn't I worry about that? Well, you see, gravity is in the downward direction and I was writing conservation of momentum in the x direction. Conservation of momentum is a vector equation. You can have conservation of momentum in the x direction. As long as Fnet in the x direction is equal to zero, you can use conservation of momentum in the x direction, and that's what we did.
You could worry about air friction, and that's a more legitimate worry, although even that, I would argue, is something that is not a big issue, because remember where conservation of momentum came from; p is force external times t. And even if there is a little bit of air friction, this is an explosion. The time interval from just before till just after, the time interval from initial to final is really short. And so, for that short period of time, the right-hand side is tiny, p is tiny, and I can approximate it as being zero, momentum is conserved. So even if there was air friction, I think this equation would still work very well.
You may have seen movies about giant asteroids heading toward the earth. It's nice to think about what physics has to say about this, instead of Hollywood. Giant asteroid heading straight towards us, this huge mass, huge velocity, big momentum, right towards the earth, bad news. So what are we going to do? The classic answer is nuking it, so you fire a nuke. Now, this little warhead, it's not very heavy, it's not going all that fast, it doesn't have much momentum. So the momentum of the total system, asteroid plus warhead, is still a big arrow pointing straight towards us. There's a lot of energy in that warhead so it blows up, but remember energy is one thing, but conservation of momentum says if you had a lot of momentum coming towards you in the beginning, you still have that exact same amount coming towards you. You may have blown it up into a bunch of little bits, but all those little bits are still heading towards the earth. Now, if you gave it enough energy, some of those bits might be heading off to the side. Conservation of momentum says that for every bit of momentum, for every amount of momentum going to the left, there will be an equal and opposite amount going to the right, because the total sideways momentum was initially zero and it's going to be finally zero. So I suppose that you could make the debris sort of go off on the sides of the earth, but pretty much, I think, it'd be a bad story.
Conservation of momentum is a principle of physics you can use in a lot of problems, all you've got to do is make sure that there's no net external force or that the time interval in your problem is so short that the external force is irrelevant. In that case, you can use conservation of momentum and solve for final conditions in terms of initial or vice versa. It's a very powerful technique.
Momentum and Its Conservation
Solving Problems Using Conservation of Momentum Page [3 of 3]

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