Program 18 - "Momentum and Conservation"

Music"What is this thing, what kind of life force is inhabiting these balls?I just don't understand this.Ohhh, this is terrible, it just doesn't fit.I don't know what to do."MusicSilico: "We're back with Science 122, The Nature of Physical Science.This is the telecourse that defines what it really means to be conservative.This is Program 18, Lesson 3.3, "Momentum and Conservation."Before we're done with this program we will have posedthe swinging ball problem and we will have revisited Newton's second law.Then we will define momentum and impulse and use thesedefinitions along with Newton's third law to define and studythe concept of conservation as it applies to momentumand to physical phenomena in general.We will see the implications and importance of conservationand its relationship to symmetry.Then we will apply the concept to circular motion before werevisit the swinging balls and pose yet another questionregarding their motion and interactions.

Here are the objectives for today's lesson.These objectives are also in the Study Guide at the beginning of the lesson.Before you begin to study the lesson take a few minutesto read the objectives and the study questions for this lesson.Look for key words and ideas as you read.Use the Study Guide and follow it as you watch the program.Be sure to read these objectivesin the Study Guide and refer to them as you study the lesson.Focussing on the learning objectives will help youto study and to understand the important concepts.Compare the objectives with the study questions for this lessonto be sure that you have the concepts under control.That's a nice voice.In fact, it's really pleasant.Why don't you use that more often?Well, hello, hi.In this program we're going to rethink Newton's lawsof motion from a slightly different perspective.And this will allow us to solve more kinds of problemsand consider even a wider variety of situations than we could before.Along the way we will encounter the concept of conservationwhich is one of the strongest and most usefulconcepts in all of physical science.

So, at this point, you may be asking why would we wantto rethink Newton's laws when there are so obviously intuitive,and works so well in gravitation and in explaining projectilemotion and all these kinds of things.Well, here's the reason why, and we'll elaborate moreon this as we go through the program.After Newton published the "Principia," physics advanced rapidly.I mean Newton had really set the stage here for not justexplaining the motions of planets, but a model for doing science,for understanding about the physical world.This whole idea of mathematical analysis and inductiveand deductive reasoning, and using universal principles and so forth.The problem was that although Newton's methods allowedanalysis of any situation involving forces and motion, that therewere some situations that couldn't be analyzed directly.

Now don't get me wrong.The whole science of mechanics and gravitation were verysound, that in fact, were very complete, and Newton's lawsare an excellent foundation for this.But, these methods of analysis could not bedirectly applied to certain types of phenomena.For example, there are situations in which collisions happenor forces are exchanged between objects where we can see theresults, in other words, when two balls collide with each other,we can see that forces are exchanged and that they doobey the laws of motion, but were unable to measure the forcesexactly, and were unable to measure the acceleration.In fact, for all practical purposes, when a baseball hits a bat,the acceleration of the baseball is instantaneous.There's no apparent acceleration involved,although we know there must be if the laws are true.So, what happened was that new methods were developed whichtook advantage of the Newtonian paradigm which allowed usto view things in a slightly different way which will enableus to see things that we couldn't see before.

Well now we come to what I like to call the swinging ball problem.We don't know if Aristotle ever had a set of this particular kindof toys, but if he did, he would have found it quite perplexing.The reason why is because there's really no explanationin Aristotle's cosmology or his explanation for motion asto why the balls should behave the way they do.But, before we start talking about the way the balls behave,I want to show you exactly what they do and then we'll comeback and try to explain this motion, or at least understandthis motion in terms of Newton's laws.This will take us a little bit of time to do this becauseit's not as easy as simply applying "F" equals "M-A"and looking at the first and second law.But, we'll come back to do that.

OK, so let's see what the balls actually do.What you'll notice here is if I pull one of the balls aside and startit swinging, that it contacts the other balls and as a resultone ball in sends another ball out on the other side.And, in fact, this makes no difference which side I start them on, right?If I do it on this side, one ball in, one ball goes out.So, what do you think would happen if I was to send two balls in and, instead?What's going to happen?Well, let's see.You see what happens is that two balls go in and two balls come out.The ball in the center is sort of a neutral observer and sort of sits there.You will notice also that the balls begin to swingin unison as they sort of wind down.We'll get into what happens there in the next program.So, what do you suppose would happen if I sent three balls in?You notice that is a total of five balls, so what happens if I send three balls?Let's see.

OK, I start it; send three balls in.You notice once again that three balls go in and three balls go out.In this case, the center ball just sort of behaves as if it's a pendulum.It sort of swinging back and forth.What do you think would happen with four balls?Is there a principle here?Let's see.Make a prediction for yourself.What's going to happen if I do it with four balls?It's a little harder to get a hold of, but...Once again you see four balls go in and four balls go out.So, is this a general principle or is this just work for the four balls?For example, what might happen if I send one ballin and one ball in the other side?What's going to happen?You see, it's still the same, same thing, isn't it?One ball goes in and one ball comes the other side.And the effect of this is that the two balls really bounce off of each other.What's really happening here is this ball's going in and sendingout this ball, and the other ball's going in, sending out this ball,so that as a whole, they appear to bounce off of each other.

Now it gets interesting.Now we can have some fun with this.What happens if I send two balls in one side and one ball in the other side?Now this looks fairly complicated, doesn't it?But when you start paying attention, you see what'sreally happening here, don't you?The same rule is applied.That the same number of balls that goes in one side comes out the other side.So, if you watch this--two balls go in one side, two balls comeout the other side at the same time one ball goes in one sideand one ball goes out the other side.Now, stop and think for a minute.How much trouble would it be to choreograph this?If you had five dancers, and you say to these dancers,"OK, I want you to line up in a line.Now, two of you move over to this side and one of you move overto this side, and when I say "Go," you all come together.And now two of you are coming in this side, so you, who just camein this side, bounces off this side, and you, who's already here goes off this side..."It would be a mess to try to get people to this sort of thing.

Now, being not a dancer, it would probably be harderfor me than it would be for most people.But, look what happens here.What happens if I send three balls in on one side and one on the other?Still the same principle applies, right?The number of balls in on one side equals the number of balls out on the other side.Now is this perplexing?Is there some way to explain this.Well, let's see now if we can summarize what we've actually said here.

OK.First of all, we see that the behavior of the balls is thatthe number of balls in one side always equals thenumber of balls out the other side.This is true whether the balls are coming in both sides,or one side, or whether the are even numbers on both sides, or not.So, the question is, what accounts for the behaviorof this system of swinging balls?How can we explain this?Is it consistent with anything that we already know?Well, there are several possibilities here.One possibility is that these are intelligent balls which wereprogrammed at the factory and we've been here choreographingand rehearsing them since this morning trying to get them to behave this way.That might work in Aristotle's system, but I don't thinkit works very well in our modern paradigm.The balls probably are not intelligent.In fact, I have a feeling if you substituted anyof just regular balls, you know the ball offthe street, it would probably be able to do the same thing.So, are there special laws then which apply only to these balls?In other words, is this a particular situation in whichspecial laws apply that don't apply anywhere else?Well, that would be contradictory also to our concept of universality.Right?That the same law should apply to all objects whether or not theyhappen to be hanging from a string, or whether they happento be on the desk top here in the studio.

So, the question then comes around is that is theirbehavior consistent with Newton's laws?Well we know that Newton's laws apply to planets.We know that they apply to projectiles.So, is their behavior consistent, or is it not?And, the answer, of course, is that if they're not, then Newton'slaws are either incorrect or incomplete and we have to modify them.So, here's an example of how we use a paradigm to analyze or to ask questions.Because what we want to do here is to ask the question,how can we analyze the behavior of the balls in terms of Newton's laws?This is the paradigm, right?We've now adopted Newton's laws as a paradigm and we wantto understand how we can do this.We have some problems with this, though.For one thing, we cannot measure the forces exchanged between balls.You remember in Newton's laws the second law says thatacceleration of the balls is equal to the force times the mass.I should say is equal to the force divided by the mass.Whoops, don't quote me on that, kids.

Acceleration is force divided by mass, so we have adifficulty exchanging the forces.We can't really attach a scale to the balls and see how muchforce they exchange with each other like we could,for example, hanging a weight over a pulleyand letting it drag a cart along.The other problem has to do with the acceleration of the balls.We can't really measure the acceleration of the balls as they're stuck.Because for all practical purposes, the velocityof the balls seems to be instantaneously acquired.Now, we know that there probably really isn't because we knowthat an instantaneously acquired velocity, accordingto the second law, would require an infinitely large force.So we don't imagine that the force is infinitely large,but what we do see is that the time intervalover which the ball changes speed is very short.In fact, it amounts to about the amount of time of a click.You can hear the click.(Click, click, click, click, click, click, click.)The amount of time over which the ball is acceleratingis about the length of that click, which is actually very short.So, what we really need to do is to revisit the second law.So, what we want to do is to revisit the second law.But we want to do this in such a way that it helpsus to understand the motion of the balls.

So let's go to the ELMO and I'll show you what Newton actually did.When Newton defined the second law, he actually did it in terms of momentum.But in our modern terminology we've come around to write it,as you remember, as "F" equals "M-A."What we want to do is to look at this and see how we canrewrite this or rethink this equation to understand it in a slightly different way.So let's go back to our definitions from Galileo of motion,and recall that by definition, acceleration is actually thechange in velocity divided by time.OK.This is Galileo's original definition of acceleration.So, what happens now if we take this termfor acceleration and put it in here in the "F" equals "M-A."Let's see what happens.So, what we write here is "F" equals "M" times acceleration,but we're going to write acceleration as change in velocity over time.OK.This is in parentheses, but those of you who are algebraicallyinclined may recall that if you have somethingin the parenthesesyou can distribute what'sin the parentheses outside the parentheses.What that means in plain English is that I can also write this as"F" equals "M" times "Delta V" divided by "T."Now again, if this troubles you algebraically, don't worry too much about it.You're not going to have to derive this equation,and the purpose of the equation, again, is to seethat there are different ways to look at this relationship.So, here's what we want to do.I want to multiply both sides of this equation by "T."So that the "T" moves over here, and multiplied by the "F"and on the right hand side, I have "M" times "Delta V."

Now, this change in velocity, remember, simply representsthe, a change in velocity from one time to another.So, I think it's also clear that you could also writethis as a change in mass or a change in velocity.Here's what I mean by that.Suppose we define the term of mass times velocity to mean something.In other words, let's define mass times velocity and call it whatNewton called it which was the quantity of motion.So what we have here now is a statement that says "F" times"T" is equal to "M" times "Delta V."So, we want to go back in a minute and examine this relationship.And we'll look at the quantity of motionand this part which Newton called impulse.But for now we want to try to understand something morebasic, and that is, why bother to go back and redefine this.I mean this seems like an awful lot of trouble.So what we're going to do now is to take a look at why do wewant to go back and reexamine the second law.So the answer to the question, "Why view the second lawin a different way," is a very simple one.The answer is "Why, not?"After all, the law is fundamental.Or at least if it isn't fundamental, it's not a good lawand if it is a good law, then it's fundamental.So, that means that the law should apply regardless of howwe look at it or what perspective we take to look at it.This is one of the nice things about our paradigm, you see.Just because the law is stated in a certain way doesn't meanthat's the only way we can look at it.So, I mentioned earlier that certain situations do not allowdetailed knowledge of either the forces or the acceleration,but maybe we can look at this in terms of change in velocity.

Now to do this I had to set up a demo.I'm sorry, my silicon friend, but I'm going to have to move youoff the tabletop because I need the space.Silico: "Oh, no, not again.I don't like sleeping in the middle of the program.What if I miss something that is important?"OK, OK, I won't put you to sleep.How about if I just move you over here like this,and I'll put these over here and you can play with them while we talk.How's that?OK?Don't make too much noise now, but have fun.OK.So, what I want to do now is bring back our carts.You remember these carts from a previous program.Now the carts actually are a model and they're a modelfor anything that's colliding.Could be the swinging balls, it could be the ballsrolling on the tabletop, it could be any kind of collision.Now you may wonder why we focus so much on collisions.And again, the idea of collisions is just a model for somethingwhere forces are exchanged, obviously, when the cartscollide, but it's very difficult for us to measurethe duration of the force and it's also very difficultfor us to measure the strength of the force.

Now don't get me wrong, this can be done.But in Newton's time it couldn't.And in trying to analyze these type of situations it wasimportant to come back and try to figure this out.So, notice what happens.One car's sitting still and the other runs into it.Watch what happens to the two cars.You notice that the cart that's moving when it contacts thecart that's standing still, virtually comes to a restand transforms all of its motion into the motion of the other cart.Let me try this one more time.See what happens there?This cart is moving, it collides with the second cart,the second cart is now moving, the first cart is standing still.One more time.OK.Now you can see that this situation involves the transferof something from one cart to the other.Now what happens if we weight down this cart.I think I had some weights here; hold on a second.Yep, there we go.What happens if I put a weight onto to this cart?You can't tell this from where you are, but each of the carts hereweighs about one, has a mass of about one kilogram.In other words, it's about the mass of one of these weights.So you'll notice now that if I weight down one cartand allow it to collide with the second while it'smoving, that something different happens.

Now the first cart continues in motion.So what happens if I weight down the non-moving cart, the stationary cart?Let's see what happens now.You notice that now the second cart, although it didn't, wasn'tso apparent the lighter cart actually rebounds.I said the lighter cart actually rebounds.Maybe I need to have more weight on here to illustrate this.Let's weight this thing down really good and see what happens.OK.So, now I've got a couple of kilograms of mass on there, let's see what happens.I hope this works.OK, so now you see that the lighter cart actually rebounds.So we have three different kinds of situations here.We have a situation when the mass of the cart is about the same.We have a situation when the mass of the stationary cart issmall and the mass of the moving cart is large.We have a situation where they're reversed.Where the mass of the stationary cart is largeand the mass of the moving cart is small.Now these all are similar situations, but they allare different at the same time.They have something in common but they something different.

Now, just a reminder of this, I want to go back for a minuteand try something different which we looked at when we studiedNewton's laws in the first place back in an earlier program,and that is, the fact that the carts exert forces on each other.Now this is true whether or not the carts are colliding thisway, or whether the carts are compressed togetherwith the spring and then they rebound.So you'll notice here that if the two carts...if I squeeze thistogether so that the spring is ready to exert a forceon both carts, that the two carts accelerate awayfrom each other; that is, cart "A" exerts a force on cart "B"and cart "B" exerts a force on cart "A."And again, if I weight down the carts, you can see thatwhichever cart has the most weight on it or has the greatestmass, has the smallest acceleration.So if I move the weights over this cart and try squeezing themtogether and letting go, you'll now notice that again the lightweight cart is the one that moves and the heavy weight cartmoves with much less acceleration and much less velocity.

Now I want to focus here, not on the conceptof acceleration, but on the concept of velocity.Now, again, it's obvious that the carts are accelerating,but what we notice about them more than theiracceleration is their final velocity.That this cart has a greater velocity than this cart,and when I switch the weights around, now once again,the cart with the greatest mass has the smallest total velocity.So there seems to be some relationship herebetween the change in velocity of eachof the carts and the mass of the two carts.So, now we're ready to go back and define what we meanby momentum and impulse and that will eventually lead usinto the concept of conservation and momentum.At last we're ready to come back now and defineexactly what we mean by momentum.Newton defined what he called his "quantityof motion" which is mass times velocity.In modern terms we simply call this momentum.The equation on the screen says "P" equals "M-V."Now, don't be confused by this.We'd like to call momentum "M" but "M's" already used for mass.So there's only a certain number of letters in the alphabetso physicists for some reason chose "P."So here we have the expression momentum equals mass times velocity.OK.Chee, you know, I just realized that I'm really alone out hereand I need my silicon friend, so let me bring him back here.Its got all my brains and all my notes and everything on it.So, we'll put this back here, oh, I feel much more comfortable now.So how was that?Did you have fun over there playing with the ball?Silico: "The balls are not much fun.They never do anything different."Well, of course they don't.That's exactly the point.OK.

So, now we can look at what is happening here.Newton defined the "quantity of motion" which we callmomentum as the product of the mass times velocity.Why did he do this?What do we talk about when the "quantity of motion."Well, it has to do with the fact that if you have somethingthat's very large, but moving slow, it still takes a lot of something to stop it.For example, consider a big battleship, you know,sitting at the pier and it's moving back and forth veryslowly with the surge of the waves.Now if you happen to get caughtbetween the battleship and the pier, it's going to takea lot of force, much more than you can come up with,to stop that battleship even though it's moving very, very slow.In fact, a battleship weighing 20 or 30 thousand tons that'smoving at only a fraction of a centimeter per second is stillsomething that you're not going to be able to exert enough force to stop.On the other hand, something very, very small, like a bullet,very small mass, but moving at a very, very, very high rateof speed also requires a lot of "something" to stop it.Now is it just force that's required to stop it?Or is it something else?Well, let's think about this.Force, we know, causes a mass to accelerate at a certain rate.So, a large mass with a small force supply willaccelerate at a certain rate.A small mass with the same force supply will accelerate at agreater rate, meaning that it will slow down faster.

So, remember now, that acceleration is the change in velocity.So what we're looking at here is how much, what's therelationship, I should say, between the forceand the amount of change of speed for a given object of a certain mass.Now I know that sounds confusing, but that's whywe want to put it in these particular terms.So, the question comes up, then, that if a large mass movingat a small speed might require the same "something" which ismore than just force to stop it, than a small mass movingat a large speed, what is that "something?"The answer is, that "something" is called impulse.So now it's time to consider what we mean by impulse.Impulse is related to momentum because a given changeof momentum can be caused by various combinations of forceand time in the same way that momentum, itself, can beattained by either various combinations of masses and velocities.So here you might consider, for example, the differencebetween a short duration force of high intensitylike that of a baseball hitting a bat.Or you might think of a small force applied over a long periodof time like that of a rocket ship accelerating into outer space.Here's a way to think of this.

Now let me ask you what sounds like a dumb question.It's not as dumb as it sounds because there's a point to it.But, suppose that you had to jump off a second floor of a building.Which would you rather land on, a hard concrete flooron the bottom or a soft fluffy mattress?Well, if you choose the hard concrete floor you're probably pretty strange.Most people would choose the soft fluffy mattress.What is it about the soft fluffy mattress that makes it more desirable?Of course it's soft and fluffy, but there's more to it than just that.For one thing when you fall, you are acquiring momentum.Right?You have a fixed mass, your velocity is changing.So, you are acquiring momentum as a resultof the gravitational force being applied to you.You are accelerating at the rate of ten meters per second squared.OK.When you stop, when you hit the ground, would you ratherchange your momentum rapidly, or would you rather change it slowly?Well, here's where we get back to the concrete versus the mat.The concrete will stop you in a short amount of time.That means that it must use a large force to do that.On the other hand, the mat will lengthen the time over whichyour momentum changes so that the force that's supplied to youto stop your motion can be smaller.So what's the effect of a large force acting on the human body?Well, of course, it has to do with the abilityof your bones and your joints and your muscles and yourbrains and so forth to withstand those forces.So, if you stop (smack) really fast like this, the acceleration thatyou undergo, is very large because a large force is required.On the other hand, if you stop slowly (smaack)over a longer period of time, your acceleration isslower because the force that's supplied is less.So, the same thing applies, for example, with paddeddash board in cars, or air bags.

All of these things have the same effect of changing yourmomentum over a longer period of time which reduces the forcenecessary to stop you, which, of course, also reduces the forcethat exerted on you and in this action-reaction,the force that's exerted on you and the force that you exerton whatever it is that's stopping you.What exactly is conservation?Does this mean that you shouldn't use too muchmomentum in the same way we say you shouldn't usetoo much water in water conservation?No, not really.We're looking at a different use of the word, conservation, here.In fact, the law of conservation and momentum was oneof the first of many conservation laws in the physical sciences.It's a way of looking at constancy amid change.What we mean by that is this.Well, actually it's a type of symmetry.Remember symmetry had to do with constancy when youchange something, the circle, for example, doesn't change when you rotate it.So here we have a little bit different meaning of the word, symmetry.But here we have something, momentum, which remains thesame while something else is changing.And it's not really anything new, it's just, again, a very uniqueand very different way to look at this.

Now, I want to recall for you that Newton's laws are still valid.And that we're simply seeing things from a differentperspective here, and you may be wondering already,why do we go to all this trouble?I think now's the time to see this.So, what really happens here is to derive this idea of conservation momentum.We want to go back and view not only the second law,but the relationship between the second and the third laws in a different way.So, let's go to the ELMO and I'll show you what I mean.OK.So here we had the last time, we had this relationship wherewe had used the second law to show that impulse equals the,what we call the "quantity of motion."So now we can write this and we can not think of thisnecessarily as "quantity of motion," but we can thinkof it as impulse equals the change in momentum.This is what we came up with during the last section.Let's see now how this applies to a collision,or how it applies to a reaction between two objects.Let's have a nice clean sheet of paper here.And now we want to go back to our Newtonian universe.Remember in the program on Newton's laws we talkedabout suppose, imagines you were a god and you can invent your own universe.

So let's go back to that universe again usingNewton's laws and start a ball in motion.So, here we have a ball.This looks very much like what we did before.A ball, let's give it a certain mass.And let's give it a certain velocity in this direction.And remember now, that in the absence of any other objectin this Newtonian universe, that this ball will continue alongon an inertial path in a straight line at a constant speed.So we're going to send a second ball of equal massaimed at the first ball at identical speeds so thatthe two balls have the same speed and we're goingto arrange this so that the two balls collide headon.Now we already know what happens.We've seen this with the case of the swinging balls, but we'vealso seen it in case of the actual balls or the carts for that matter.Keep in mind this, this doesn't only apply to balls,it applies to anything which is going to collide.So after they collide, here's what happens.This ball's moving off in the opposite directionand its counterpart is also moving off in the opposite direction.Both balls have changed directions.

OK, what we want to do now is to look at the interaction between the two balls.So what actually happens up here when the two balls collide?I'm sort of drawing them a little smaller.What actually happens at the time they collide?Well, we saw from the third law that the two ballsactually exert forces on each other.So let's examine those forces, but let's examine them nowin terms of our relationships of impulse and momentum,rather than in terms of simply mass acceleration.So let's look at the forces on the balls.Let's again analyze first of all the force on this ball.We understand that it has changed momentum.Right, it's changed velocity.In fact, it was formally moving in this direction,it's moving in this direction now, so we can say thatthere must have been a change in velocity whichin this case involves a complete turn around in direction.You might notice here that if we multiply the mass of that balltimes its velocity we have here a change in momentum.Let's do the same thing for the other ball.So we look at this ball, and we'll see that this ball also has acertain mass and changes its velocity.What about the direction of the change of velocity?Notice that this ball was moving to the right.Now it's moving to the left.This ball was moving to the left.Now it's moving to the right.So, whatever the change in velocity is it's opposite for the two balls.

In a typical mathematical coordinate system we wouldsay that this direction is positive and this direction is negative.Although it doesn't make any difference which one you callpositive and which one you call negative, as long as yourecognize that they're opposite, so one is positive and one is negative. So, the idea here is that this ball has changed momentum.Let's call this, in fact, a negative momentum change, and this ballhas changed momentum and let's call it a positive momentum change.This comes from the fact that the change in velocity is positivein one case and negative in the other case.OK, so we see now that each ball has changed velocityand, therefore, has changed momentum.With the plus and minus sign we can also look at it this way.We can say, this ball has a positive change in momentum;this ball has a negative change in momentum.So, we can look at this and say, a positive change in momentumcan be construed as a gain in momentum.On the other hand, we can say that a negative change in momentumcan be construed as a loss of momentum.We'll come back and look at this in a couple of minutesin accounting terms rather than terms of balls.So, for now just think of one of the balls gainsmomentum, the other ball loses momentum.

Now the question is, how does the gain of momentum of one ballcompare in size with the loss of momentum against the other?I'm about to show you here that if you believe in Newton's laws,that the loss of momentum of one ball has to be exactly equalto the gain of momentum of the other ball.And here's why.We know from our relationship that this change in momentummust be equal to the force times the time.That's the definition of impulse.And by the same hand, on the same line of reasoning, the changein momentum of this ball must be equal to the force times the time.So here's the question.How do the times of the two balls compare?I think it's pretty obvious, isn't it?That the two balls are in contact with each other for the same amount of time.But however long ball one is in contact with ball two,ball two is in contact with ball one for the same amount of time.So, the times here have to be equal.So this time has to equal this time.What about the forces?

The third law guarantees us that the forces exchanged are equal and opposite.So that means that this force has to equal this force.Specifically, they're equal and opposites, so, this force is anegative and this force is a positive.Then again, it doesn't make any difference which way we lookat the forces but I'm taking the same conventionfor the direction that I used here for the change of momentum.So, what does that mean?Look at these two equations.Oops, did I use the word equation?What I mean, of course, is relationship.Look at these two relationships.Here you have minus "F-T" equals minus "M Delta V."Here you have plus "M Delta V" equals plus "F-T."They're the same thing aren't they?In fact, if you multiply both sides of this relationshipby minus one, these become pluses.So what does this all mean?What it means is that the loss of momentum by one ball exactlyequals the gain of momentum by another ball, or the other ball.In other words, one ball loses momentum and the other ballgains momentum and they do so in exactly the same amounts.The loss of momentum of one ball is exactly equalto the gain of momentum by the other ball.Now that makes sense, I think, from this logic.

So, now let's take this one step further and say this.It is as if the momentum of one ball has been transferredto the other, because one ball loses, another ball gains.The result of the interaction is that the balls change direction.There are forces exchanged.There's a velocity change, but something remains constant here.What remains constant is the total amountof momentum shared by the two balls.OK, let's review this logic.Let's go back and take a look at the logic and then we'll comeback and look a little bit at some more implications of this.OK.So, here we see that the forces on the balls are equal and oppositeand also they exist for the same duration on both of the balls.So that means that the impulse on the balls must be equal and opposite.If the impulse on the balls is equal and opposite,then that means by definition that themomentum change must also be equal and opposite.Because impulse equals change in momentum.So, if the equal and opposite momentums change means thatone object loses and the other object gains in equal amounts.The loss of momentum by one object exactly equalsthe gain of momentum by the other object.So if one loses and the other gains the same amount, then the totalamount shared by the two balls remains constant.

So now we can explicitly state what we meanby the law of conservation of momentum.What we're saying here is the total amount of momentumin a closed system remains constant, but may betransferred from one object to another.If the total amount remains constant,that it may be transferred from one object to another.Another way of saying this is that the totalmomentum before equals the total momentum after.Before and after what?Well in this case it's after the collision.But in general we mean before and after any typeof interaction where forces are exchanged.So I want to show you this in mathematical form, and again don't panic.You know, if you're not up to the mathematics, but lookat how the symbolism works here.We had two objects, right?We had one mass and a second mass.They each had some initial velocity.The zero here simply means initial velocity.So, if we take the momentum of one object, add to it themomentum of the other object.Again, remember that this is a number, mass times velocity is a number.So, the momentum of one object before plus the momentumof the object before the collision equals the final momentumof the one object, plus the final momentum of the other object.

Now I note here that we're using plus signs here,but remember that the plus signs take into account the direction.So that when the two balls are moving in opposite directions,one has a plus, a positive velocity; the other has a minus velocity.Now this is how we would do this for these two ballswhere we have two objects involved.But in general, we can simply say it this way.That the sum, you remember this symbol from the netforce in Newton's vector algebra.The sum of all the initial momentums equalsthe sum of all the final momentums.So in general, the sum of all the momentums before the collisionequals the sum of all the momentums after the collision.Now, this might be a little hard to understandbecause momentum is kind of a new concept.So, before we go any further with this, let's try to put thisinto perspective of something we all know something about.Let's talk about money.

OK, now here is how it works with money.Suppose we consider two people, Jill and Bill.When they meet Jill has no money and Bill has $10.So the total money between Jill and Bill is $10, because zerofor Jill plus $10 for bill, the sum is $10.So, suppose now that Bill gives Jill $5.Or if you prefer, you could say, that they start outthe other way that Jill has the money.I don't care who has the money.But Bill gives Jill $5.So, now the sum is $5 for Bill, $5 for Jill, how muchmoney do they have between them?They still have $10, right?So, the sum of money is not affected by the interaction.What's affected is who has the money.In fact, anything that Bill gives Jill or Jill gives Bill,any exchanges of money that they make back and forth, the totalamount of money between the two of them is $10.And as long as there's no one else involved,then the total amount of money still stays at $10.The loss of one equals the gain by the other.OK.It's a closed system.If neither Bill nor Jill acquire moneyfrom outside or lose money to the outside.OK.

Let's go see how this works with momentum.Let's go to the ELMO.So, suppose we have a couple of carts.Remember the carts, and suppose now that one cart,here's a cart, has wheels on it.Suppose that this cart is moving with a velocity such that itsmomentum is ten Newton seconds.Oh, I just introduced a unit here.Newton seconds is a unit of force multiplied by unit of time.And since that's equivalent to a unit of momentum,we can talk about momentum in Newton seconds which,after all, is a much more better unit to talk aboutthan kilogram meters per second squared.But you might want to look at the units in the textbook to see how this works.So, this cart is moving, it has momentum of 10 Newtonseconds and it runs into another cart which at first it's at zero.It's at rest.It's at rest, it has zero momentum.OK.So, what happens when the two carts collide.Well, we saw on the tabletop what happened, right?So, here's the situation before.The total momentum, what I called summarizationof "P-0" before is equal to 10 Newton seconds.Afterwards, what happens?Well, we saw before that if the masses of the carts are thesame, that this cart now goes off with momentum of 10 Newtonseconds and this cart comes to rest.OK, its momentum is now zero.The total momentum distributed between the two carts,the final momentum, the summation of the finalmomentums is still equal to 10 Newton seconds.So, what's actually happened here?Well, there are many different ways to look at this.But, one thing you can see is that, yes, sure, one cart ismoving before and the other is moving after, but what you cansee here is that this cart has transferred its momentum to the second cart.

A transfer of momentum is taking place muchin the same way that the transfer of money hastaken place between Jilland Bill, and what wecan say here is that if the masses of the carts arethe same, then the final velocity of this cartmust be the same as the initial velocity of thiscart, if all the momentum is transferred.So here we have a way to predict how, if a momentum istransferred, in other words, if this cart comes to restat the end, we can predict that the speed of this cart will be thesame as the speed of this cart was because it now contains allthe momentum and has the same mass.The point of this is, and the beauty of this, I guess you couldsay, is that the carts don't have to have the same mass.Because once we know that the momentum transfer is,takes place, that one cart loses momentum and the other gainsit, then we can predict what the final velocity of the second cartwould be whether or not it has the same mass as the first one.There are two other quick examples that I wantto illustrate that do not involve collisionsor at least different kinds of collisions.The first of these is the recoil of a gun.This can also be explained in terms of Newton's third lawwhich we did before, that the gun exerts a force on the bulletand the bullet exerts a force on the gun.

We can also think of this in terms of conservation and momentum.Here for example we have cannon sitting with the cannon ball inside.The sum of the momentum is zero.Neither the cannon nor the cannon ball is moving and so the totalmomentum of the system is zero.After the cannon ball is fired, the cannon ball which has arelatively small mass comes out of the cannon at a relatively large velocity.I've a small "M" and a large "V" here to illustrate this.At the same time, the cannon which has a relatively largemass recoils backwards with a relatively small velocity.So, notice here the direction.There's a negative sign attached to the momentumof the cannon because its moving to the left.There's a positive sign attached to the momentumof the ball because it's moving to the right.But still, the sum of the momentums is equal to zero.In other words, the mass of the cannon times its velocity isexactly equal and opposite to the mass of the cannon ball times its velocity.So, momentum before the gun is fired is zero.The momentum after the gun is fired is zero.The second example is what's known as a ballistic cart.In the early days before there was strobephotographs this was away that was usedto measure the velocityof something moving very fast, like a bullet.The way it works is very simple.You have a rather large object here.In this case it's a wooden cart with a large mass.I've indicated that with a big "M."It's sitting still.It's on nearly frictionless wheels, sort of like thecarts that we used here in the demo.

The bullet has a small mass but it's coming at the cart with a large velocity.So, afterwards, the momentum of the cart and the bullet is combined.I should say that the bullet sticks into the cart.The cart is now moving away at some velocity, but becauseof the large mass of the cart, the combination of the bulletand cart moves at a slower velocity than the bullet was moving.So, if you know the mass of the bullet, if you knowthe mass of the cart, and if you know the finalvelocity of the bullet plus the cart, you can set up anequation to say that the total momentum herewhich is "M" times big "V" must be equal to themomentum at the end which is big "M" plus little "M," the quantity, times little "V."So you can use this then to calculate the originalspeed of the bullet even without having any otherway of knowing what its speed might be.So before we go on with this I want to sort of bring us backto earth here and look at some of the implications and importanceof this idea of conservation and momentum.We don't just do this as an intellectual exercise,although it was an interesting one.For one thing, this concept of conservation of momentum isuseful in predicting motion after interaction of various types.This is usually where forces are complex or where many objects are involved.For example, a sky rocket which explodes in flight.If you add up the directions and speeds of all the little pieces,it still equals the momentum that it had before it exploded.Another use is in car crashes.

The police can determine how fast the cars weregoing if they know the mass of the cars and thedirection that they skidded after they raninto each other, say at an intersection where they collide at right angles.The other thing, and probably more important than the actual use,was that this idea of conservation began a search for other typesof quantities like momentum which might be conserved.And, in fact, this search was so successful, as we'll seein the next program when we study conservation of energy,that we can talk about such things as conservation of angularmomentum or spin, and conservation of electriccharge and conservation of mass.So, in fact, modern physics describes the universe mostlyin terms of these conservation laws, and we will come backand see how the conservation of massenters into things in a later program.The other thing, it's sort of a Pythagorean thing, that thereis this relationship between conservation and symmetry.Because, after all, conservation, remember,means something that's changed upon interaction.Here we have in this midst of chaos of an explosion likea fire, a sky rocket going off, or a collision or in the swingingballs, with all these things going on and all these changes thatare so hard to explain, one thing remains constant.And remember that describing change has always been thegoal of physical science, ever since Aristotle's time.So, on one hand you have all these things changing in chaotic sortof way, but in this midst of chaos, one thing remains constant.That one thing in this case is momentum.

Now it's time to turn our attention to one application of the idea of momentum.This has to do with what we call angular momentum.When we're talking about angular momentum we really mean circular motion.The word, angular, comes from the idea that when something'smoving in a circle, it's constantly increasing its angle.We want to define angular momentum as simply linear momentum.That's mass times velocity times the radius of the circle.And when we say now that angular momentum isconserved, let's see what that means.Suppose you have something moving in a circle.If the mass of that object remains constant, in other words,if it doesn't eject any of its particles or any of its mass,then in order to keep angular momentum constant,if the radius of its orbit changes, in other words, if it movesfrom a large diameter to a small diameter orbit, in order to keepangular momentum conserved, its velocity must increase.We'll come back after the demo and show you how this works.But, there's not enough room to do this here, so I think, I bettergo to the big demo area to do this.

Well, here we are in the big demo area.Oh wait a minute.I was going to do this thing with the bicycle wheel, but I forgot the wheel.Oh, there it is!This is a bicycle wheel.Actually it's an ordinary bicycle wheel except that it's gothandles on it and it's not attached to a bicycle.Did you ever wonder what keeps you up when you're riding a bicycle?You probably know that when you move it's much easier to stayupright than it is when you're not moving.Let's see what happens and why that works.Notice that I left this hang.It hangs vertically.That the string is perfectly straight down, the wheel's not turning.What happens if I start the wheel turning?Let me give it a good spin, and hold this up like this.Look at that.Is that magic or what?You see what's happening is that the spinning of thewheel somehow holds it horizontally.This is basically what happens when you ride the bicycle.What's really going on here has to do with conservation of angular momentum.I don't want to get into the mathematics here.It's a little more complicated that what we're used to seeing,but it has to do with the fact that the wheel, itself, each pieceof rubber on the wheel is trying to go in a straight line becauseit has inertia, and at the same time it's trying to fall oversideways, and so, it follows this curved path.

You'll notice that as the wheel slows down it begins to losethis horizontal attitude and eventually will come to rest, being vertical again.So, if you're wondering what it is that keeps you up riding a bicycle, it's exactly this.I don't know if you can see this or not on the screen,but I'll show you in a minute how this workswith something else, but I'll give this a good spin.If I try to turn the wheel like this, it fights back.I don't know if you can tell that it's fighting or not, but youknow, did you ever hear anybody say they don't fly in airplanesbecause they can't see what holds them up.You don't hear that so much in the 90s any more,but people used to say that all the time.You might ask somebody like that, well, what holds you up when you ride a bicycle?And what it is, of course, is that angular momentum is being conserved.OK.I think that's enough for the wheel.Let me stop this thing.

I want to show you something else now with the weights.This is sometimes called the multidumbbell experiment.So, let's sit down on the stool.What I've got here is a stool.It's not an ordinary stool.It's actually a stool which has very low friction bearingsand a stool which has a foot rest that I can put my feet up on.Now, here's what's going to happen.Have you ever seen an ice skater who goes into a spin with herarms outstretched like this and then as she spins on her toes,she pulls her arms into her side like this and spins fasterand faster and faster and faster and then when she wantsto slow down, she puts her arms back out again?Well, this is exactly what's going to happen.So let's take a look at the variables here for a minute.We defined angular momentum as mass times velocity times radius.So what I can do with the weights is to changethe radius of their circular orbit.

Now let's see what happens.I'm going to start out with my hands out like this.Put one foot up here and give it a little bit of a spin.OK, so notice I'm spinning fairly slowly.I pull the weights in to my side.What happens?The act of pulling them in makes me spin faster.To slow down I can bring the weights back out to the side again.And I'm actually losing a little bit of my momentumbecause there's friction here with the stool, but even so,when I bring them into my side, I start to spin faster.I also get very dizzy doing this, so if I happen to fall down orsomething, don't worry about it, it happens all the time.I'll try this one more time, just so you can see what happens.OK.Here's the long, the big radius.OK.Get spinning, so now I'm shortening the radius.And as a result, the velocity speeds up because theangular momentum remains constant.To slow down, I send the weights out like this again.Whew, I really am dizzy.Is that the camera?Yeah, I think so.OK.I want to show you one more thing here now with the bicycle wheel.If I'm sitting here with my feet up, off the floor and try to movethe bicycle wheel, I can do this with no problem whatsoever.What happens if I try to do this while the wheel's spinning?Let's see what happens.I'm going to get the wheel spinning like this, give it a pretty good spin.Watch this.As I tilt, the act of tilting the wheel causes me to turn on the stool.In other words, the force that I'm exerting to twist the wheelalso results in a torque that twist the stool.This can never happen if this wheel's standing still.In fact, I can sit here all day tilting the wheel backand forth, nothing of the kind happens.So here again we see an example of how the force that I hadto exert to overcome the angular momentum of the wheel istranslated into the movement of the stool.Well I think that's about it for this demo.

Let's go back to the regular studio.Oh boy, that was fun.Still a little bit dizzy from that.You know this idea of angular momentum is really animportant one and it applies in many different areas.We can look at Kepler's laws, for example, and see that when theplanet is further away from the sun, it has to slowdown because its velocity changes.In fact, let's look for a minute at the equation, at the relationship.Here's the definition of angular momentum.Again, we call it "L" simply because all the other letters were taken up.But it's equal to linear momentum which is masstimes velocity times the radius.So, if the mass remains constant, as it didin the case of the weights, then the variables are thevelocity and the radius of the circle.So, if the radius of the circle decreases, then the velocitymust increase in order to keep the angular momentum constant.So in the case of Kepler's laws, if the planet is fartherfrom the sun than "R" is large, so its "V" must be small.And if the planet is close to the sun, "R" is small, so "V" must be larger.There's one more example I want to bring you to with this.This has to do with cats landing on their feet.You've heard this, right, cats land on their feet.

Now you can try this at home but you know be nice to your feline,don't drop her off of the 14th floor of the condo or anything.If you're really going to try this, do it over a bed and don't let herfall too far because we don't want to be accused of cruelty to animals.So, you have to do this in the right sense.But if you take a cat and hold the cat upside down like thisand leg go of her or him, as the case may be, sure enough thecat will on its way down to the ground, turn around.You gotta ask, how does the cat do this?Because in order to twist, in other words, to change yourorientation in space you need something to push against.Right?Because you have to exert force on something and if there's nothingthere to exert force on, then how do you do it?Cats have this ability, genetically programmed, to do this usingconservation of angular momentum.The way they do it is quite interesting.Cats, like the rest of us, have four limbs, right?So, if you drop the cat, what happens is that the cat willtake its front legs and put them close together like this,while spreading out the back legs like this.So that means that the back legs are harder to turn than thefront legs, so the cat turns the front half of the body aroundusing the back legs as something to push against.Once half of the body is spun around, then the opposite happens.Now, the front legs splay out like this, the back legs cometogether like this, and so this half, the front half of the bodynow becomes more hard to turn, so the back half now spinsaround and the cat lands on its feet.So it's a two stage process.The front half turns using the back half as harder to twist.The back half turns around, the cat lands on its feet.I'd say that's pretty amazing, wouldn't you?

Now it's time to go back and revisit the swinging balls.You remember the swinging balls, don't you?That's what we started out the program with.And it was understanding the behavior of the swinging ballsthat got us involved in all this stuff about momentum and conservation anyway.So the idea is that the same number of balls go in as come out.What we want to ask is, is that the only way that this can happen?Let's go to the screen again and see what this looks like.Here we had the situation.We have five balls, one of them's moving, the other four are not.The one that's moving has momentum equal to "M-V," the rest have zero.So the total momentum before the collision isequal to the momentum of one ball.That's "M-V."After the collision, one ball is moving.The total momentum after the collision equals "M-V"and we have fairly good faith that if the masses of the balls arethe same, the speed of this ball afterwards is the same as this ball before.The questions is, is this sufficient to explain the behavior of the balls?I think we can find a situation where momentum is conserved,but where something else happens.Here, for example.Suppose two balls were moving in.

Now the total momentum is two balls of mass "M" so that'stwo "M" times their velocity, "V," which equals two "M-V."So, momentum before the collision equals two"M-V."Now afterwards, suppose that one ball came out moving at twice the speed.Now you have one ball of mass "M" moving at twice the velocity.It's still two "M-V,"So, the momentum before and after the collisionis the same, but here you have one ball going in, twoballs going in, sorry, and one ball coming out.The question is, why doesn't this happen if bothsituations conserve momentum?In fact, there are many situations you canimagine which conserve momentum.Here's another one, for example.You might, for example, have two balls going in at a certainspeed, and four balls coming at half the speed.Still the momentum is conserved.Before it's two "M" times "V" which gives amomentum before as two "M-V."We have afterwards momentum, four ballscoming out at half the speed.It's still four masses times one half "V" equals two"M-V."So the question is, if this is not sufficient, is theresomething else that's required.Momentum, itself, although it's a necessary condition,doesn't seem to be a sufficient condition.So the question is, what else is required?

Well, here we are at the end of the program again.This program we have covered a lot of ground.We've seen how momentum is defined.We've seen how momentum is conserved.We've seen the importance of conservation.We've seen how it relates to angular motion as well aslinear motion, but that's about all we can say about momentum at this point.In the next program we'll look at the necessary and sufficientconditions which will conservation of energy.I guess that's about as much as I can say this time, except that,remember, when it comes to science, get physical.You've been awfully quiet.Are you still mad at me because I made you go away?Ah, the silent treatment.I guess that's going to happen.You'll be talking later on tonight when you wantto watch television, you wait and see.Music