Since the letter "m" already represents mass, we can t use it for momentum. So someone came up with the lower case "p" as the symbol for momentum.

Transcription

1 Physics 601 Momentum Why does a gun kick when it s fired? And how does the answer to this question also explain how a rocket ship is launched into space? Why does a golfer follow through to drive the ball a great distance? And how does this answer explain how air bags protect us during a car crash? It all has to do with our topic for today, momentum. (Read objectives on screen.) We ll answer the questions about the golf swing, the gun kicking, and the air bag during this program. But let s start with some more basic questions. First, let s say that you want to push a parked vehicle a few meters. Which would be harder to start moving, a little sports car or a big pick-up truck? And which one-word physics concept is the reason? I hope you said that the more massive truck would be harder to start moving because it has more inertia. So far, so good. Now, here s the next question. Which would be harder to stop moving, a slowly rolling truck or a sports car running at the speed limit? That s a little harder to answer, isn t it? And it s because we ve added motion to the picture. We need a new term for moving inertia, and that is momentum. Momentum is sometimes called moving inertia. Since the letter "m" already represents mass, we can t use it for momentum. So someone came up with the lower case "p" as the symbol for momentum. The equation for momentum is p equals m times v, or mass times velocity. Mass is the inertia part of momentum. And velocity is the in motion part. So momentum is inertia in motion. The unit for momentum is derived from the MKS units for mass and velocity. The MKS unit for mass is the kilogram and for velocity it is the meters per second. So the unit for momentum is the kilogram meter per second. No one has come up with a nickname for this unit like they did with the newton, so we just say it the long way. The momentum equation is so simple that using it to solve problems is just a matter of plug and chug. In fact, we have so much confidence in your ability to do this on your own, we aren t even going to show you any examples. Your local teacher will give you some problems to try, and we ll just go over the answers when you re finished. Local Teachers, turn off the tape and give students problem set number 1 from facilitator's guide. (Pause Tape Now graphic) (text on screen)

2 Part a of number one is just plug and chug. The momentum is 2,500 kilograms times 18 meters per second, or 45,000 kilogram meters per second. There s no need to start part b if you understand what momentum is. If an object is not moving, it has no momentum. So the answer is zero. In number two a, the momentum of the baseball is 4.7 kg m/s. For part b, we rearrange the momentum equation to get v equals p divided by m. When we plug in 4.7 kg m/s for momentum and 7.3 kg for mass, we get a velocity of 0.64 m/s. Those were easy, weren t they? Just don t forget that an object at rest has no momentum. Now back to some more questions. What do you think it takes to change the momentum of an object? Tell your teacher. If you said a force, you re partly right, and if you said a net external force, you re getting closer still. You re using what you already know and that s good, but you re thinking about acceleration, or changing the velocity of an object. Actually, there s more to momentum than just velocity, so we need more than a force to change it. Put your pencils down and watch this. We ll take notes in a minute. ( F=ma on screen) Let s start with something you re very familiar with, the mathematical equation for Newton s Second Law. Now let s substitute the definition of acceleration, which is change in velocity divided by time. That puts the v in our equation. Next, we rearrange the equation to get everything on one line to get force times time equals mass times change in velocity. We re almost ready to get this important equation in your notes, but we have one little change to make first. Since changing the velocity of an object changes it momentum, let s temporarily move the change in sign to put it in front of the m times v. We ll put it back later. Now we re ready for notes. This important equation is called the impulse equation. A force times a time interval is called an impulse. And you already know that mass times velocity is momentum. 2

3 It takes a net external force acting over a certain time period to change the momentum of an object. In other words, an impulse is required to change momentum. And vice versa. A change in momentum creates an impulse. Now, we re not going to do any math problems with this equation. It s a lot more fun to use it to explain all kinds of things that happen in real life. Let s start with a question we asked at the beginning of the program. (golfer on screen) Why does a golfer follow through on his swing when he s hitting a long drive? Is it to look good to the crowd or to make the ball move in a straight line? Although both of those are good reasons, it s more basic than that. Following through on a swing actually makes the ball move faster. You already know that the greater the initial velocity of a projectile, the farther it will travel as it falls to the ground. Right? But how does a follow-through increase the velocity of the ball? (equation on screen) The answer is in the equation. Following through on a swing keeps the ball in contact with the club for a greater period of time. Let s see how an increase in time can affect the velocity of the ball. (equation on screen) The impulse equation is full of variables. Some remain constant and others change. Notice that we ve put the delta sign back with the "v" since, in most cases, velocity changes while mass stays constant. And that s true in the case of the golf drive. The mass of a golf ball is constant. That s in the rule book! To keep everything straight, I m going to underline the variables that stay constant in this situation. So we have three variables that can change. Since we re concentrating on time s effect on velocity, let s not consider force for now. We ll keep it constant, too. That leaves two variables to look at: time and change in velocity. We want to see how time and velocity affect each other. In this case, we want a large change in velocity, so I ll draw an arrow up for increase. Now look at time. It s on the opposite side of the equation, so we know that "t" and delta v are directly proportional. When one increases, so does the other. So, to increase velocity, we need to exert our force for as long a time as we can. Now let s look at force, too. Everyone knows that to hit a long drive in golf or a homerun in baseball, you need to exert as much force as you can. But some students look at the equation and see that force and time are on the same side, so they conclude that if time increases, force must decrease. Not this time! Both force and time are deliberately changed by our golfer, so they are both independent variables. There can be only one responding variable in each situation, and in this situation change in velocity is it. Both force and time are manipulated to get the desired effect on velocity. They do not affect each other in this situation. To get the greatest change in velocity, we exert a large force over a large time interval, and that gives 3

4 us a really large impulse, which creates a really large change in the velocity of the ball. So following through on your swing keeps the club head or the baseball bat pushing on that ball for a fraction of a second longer. And that small increase in time makes a big difference in how far the ball travels. Just ask Tiger Woods or Barry Bonds. Now, you use the impulse equation to explain why a batter stops the bat when he bunts the ball. Talk it over and tell your teacher. Then we ll come back with more examples. (Pause Tape Now graphic) Did you say that stopping the bat decreases the time of contact with the ball and therefore decreases the velocity of the ball? I hope so. Now, let s try another situation. You are riding in a truck going 60 miles per hour, and the brakes have failed. To stop the truck, would you rather run into a brick wall or a haystack? You probably know the answer, but what if you were asked to use the impulse equation to explain? Watch and learn. (cartoons and equations on screen) In this situation, two of the variables are constant, the mass of the truck and the change in its velocity. The truck has to go from 60 miles per hour to zero in both cases. The change in momentum of the truck is constant and creates an impulse. So we re left with time and force. When the truck runs into a brick wall, the time that it takes to stop the truck is small. Since force and time are on the same side of the equal sign, they are inversely proportional. A decrease in time increases the force applied to the car. On the other hand, when the truck runs into a haystack, the haystack gives or collapses and stops the truck more slowly, increasing the time over which the force is applied. Increasing time decreases force, so less damage is done to the truck and driver. This idea of decreasing force by extending the time it takes to stop an object can explain so many things. Air bags give as you hit them to extend time and decrease the force exerted on your head. For the same reason, boxing gloves are padded and guard rails on dangerous curves are designed to bend and collapse on impact. And when you jump or fall, you are advised to bend your knees when you hit the ground. Why? Again, extending the time it takes you to stop decreases the force exerted on your bones. The person who tenses up on impact usually gets the broken bones. (football players on screen) So why does being tackled on Astroturf hurt more than on natural grass? What do you think? Tell your teacher. 4

5 It s not the grass itself, but what s underneath that s the key to the answer. Astroturf is laid on top of concrete. And concrete does not give like the ground under real grass does. That small difference in the time it takes you to stop makes a big difference in how you feel the next day!! (people holding blanket on screen) You can try this at home or outside at school. Have two people hold a sheet or blanket so that it sags. Then throw an egg at the blanket. No matter how hard you throw, the egg probably won t break. And you should be able to explain why. Your teacher has a contest involving eggs and the impulse equation that you might like to try. ( Physics Challenge on screen) Here s a physics challenge for you. A rubber ball and a ball of equal mass made of modeling clay are thrown at the same velocity at a window. Which is more likely to break the window? a. the rubber ball b. the clay ball c. neither. They will exert the same force on the window. The answer is a. The rubber ball will exert almost twice the force on the window as the clay one. That s because the rubber ball will bounce. Watch. (text on screen) Let s say that both balls have an initial velocity of five meters per second. The blue clay ball hits the window and sticks. So its change in velocity is from five meters per second to zero, for a delta v of five meters per second. But the red rubber ball hits and bounces back with the same velocity. We ll use negative five meters per second for the opposite direction. So its delta v equals five minus negative five or 10 meters per second. Twice the change in velocity means twice the force exerted on the window. (student on screen) To see the difference that bouncing makes on the force of impact on objects involved in collisions, watch this demonstration. Look at these two darts. One has a nail embedded in its tip, while the other has a rubber tip. We ll pull the nail-tipped dart back so that it is released from a certain height. Watch what happens when it collides with a wooden block. You can see that a force is exerted on the block, but not enough to knock it over. Now watch what happens when we use the rubber-tipped dart, which will bounce as it hits the block. We ll release it from the same height, so that it will be moving at the same speed as the first dart when it hits. This time the block is knocked over. 5

6 Enough about the impulse equation, already. Let s move on to another topic involving momentum. Have you ever seen a basketball player jump up and then seem to defy gravity, hanging in the air when he should be on his way down? Well, we can explain this using an important law of physics. It s called the law of conservation of momentum. Better get it in your notes. The law of conservation of momentum states that the momentum of a closed, isolated object or system of objects does not change. This means that momentum is conserved. It cannot be created or destroyed. There are three terms in this statement that need to be defined. They are closed, isolated, and system. To show you what these terms mean and how the law of conservation of momentum works, we ll use a cart and two steel balls on a slanted ramp attached to the cart. A system is a collection of objects. In this demonstration, the system is the cart and balls on it. A system is closed if no objects enter or leave. Our system is closed because no balls will be added to or removed from the cart. A system is isolated if no net external force acts on it as a whole. In this demonstration, no outside force will push or pull the cart. Watch what happens when the balls roll down the ramp to the right. The cart moves to the left. The cart was at rest and then it started to move with no external force applied. Does this violate the law of conservation of momentum? No, it doesn t. Remember that when we describe the motion of an object, we must consider the object as a single point, the center of gravity or center of mass. It is the motion of this point that must not change, no matter what the different parts of the system are doing. Watch the demonstration again, paying close attention to the center of mass of the system. (cart on screen) As the steel balls roll down and toward the back of the cart, the center of mass also shifts toward the rear of the cart. In order for the center of mass to remain in a stationary position, the cart must move slightly forward. Now let s use this idea of the center of mass to explain how a basketball player seems to hang in the air. When he jumps up at an angle, he becomes a projectile, and his center of mass must follow this trajectory unless some outside force acts on him. If he jumps up with both arms raised and at about here brings one arm down, his center of mass drops 6

7 about two inches because of what he has done internally. Since no external force has acted on him, his center of mass must stay on its trajectory. So his entire body goes up about two inches higher. It looks like he is defying the law of gravity, but his body is just obeying the law of conservation of momentum. Now let s go back to our demonstration. When the balls reach the end of the cart, the cart stops moving. We ll start there. The momentum of the system is zero. What will happen when the balls leave the cart? Watch. (cart on screen) When the balls leave the cart, they roll forward. The cart moves in the opposite direction, but not as fast as the balls move. Is momentum conserved? (diagram of cart on screen) The answer is yes. As long as no external force is exerted on the system, momentum is always conserved. When the cart and ball are at rest, it s obvious that the momentum is zero. After the balls leave the cart, the total momentum must still equal zero. The more massive cart moves to the left with a small velocity and the low mass balls move to the right with a greater velocity. The sum of the momentum of the balls and cart equals zero. We ll use real numbers later and solve problems involving this law. (cart on screen) Now watch as more balls are placed on the cart. As each ball leaves the cart, the cart gains velocity in the opposite direction. The same principle explains how a jet engine works. High speed gas molecules leaving the engine and moving backward make the plane move forward. Those of you who are physics types may be thinking that this sounds a lot like Newton s Third Law, the law of action-reaction. The truth is that the two laws explain many of the same things with different words. However, sometimes, one law is better than the other. (helicopter on screen) For example, the blades of a helicopter push down on air and the air pushes up on the helicopter. Newton s Third Law explains that best. But what about the space shuttle moving through empty space? There s no air to push back. So the Law of Conservation of Momentum works best here. Just like a jet engine, the rocket engine emits high-speed gas molecules that make the rocket change its momentum in the opposite direction. The total momentum of the rocket and gases remains constant. 7

8 The Law of Conservation of Momentum applies to objects that come apart, such as balls rolling off a cart or gas molecules spewing out of a jet engine. And it also applies to objects that collide. In our next lab, you ll be observing both explosions and collisions. Won t that be fun? We ll start with two kinds of collisions between lab carts. Now you know that in order to study momentum changes during collisions and explosions, we must again use a closed, isolated system. In this lab, the system will be the two carts. Our system is closed because no carts will be added or removed during the collisions. And once we put the carts in motion, nothing outside will interfere during the collision. So our system will be isolated. The carts will exert forces on each other, but these are internal forces between parts of the same system. In Part One, the carts will be turned so that the magnets on the ends repel each other. This will cause the carts to bounce off when they collide, like this. Collisions in which the objects bounce off each other and separate, without any lasting deformation or energy loss in the form of heat, are called elastic collisions. Collisions between billiard balls, bumper cars in an amusement park, and these metal balls are all considered to be elastic. They bounce off each other without any lasting deformation. Of course, these are not perfectly elastic collisions, because there is a small amount of energy lost as heat. The only particles we know that have perfectly elastic collisions are the molecules of a gas. When the molecules collide with each other and the sides of their container, they bounce off each other with the same amount of energy they had before the collision. Otherwise, gas molecules, like the air we breathe, would slow down and turn into a liquid. and we wouldn t want that, would we? So, when we talk about elastic collisions between carts, bumper cars, and balls, remember that we re not claiming them to be perfect. In all elastic collisions, perfect or not, momentum is conserved. In Part Two of our next lab, the carts will be turned so that the Velcro ends face each other, causing the carts to stick together when they collide, like this. Collisions in which objects stick together or are deformed in some way and lose energy as heat are called inelastic collisions. Collisions between freight cars that link together, between a car and tree, and between a football player and the runner he tackles are all examples of inelastic collisions. During these collisions, the objects stick together or are deformed, and energy is lost to the surroundings in the form of heat. Even when a collision is inelastic, momentum is still conserved. Momentum is conserved in all kinds of collisions, no matter what happens to energy. You ll see all that in the lab. We re almost out of time for this program, so we ll start the lab at the beginning of our next program. 8

9 But for now, it s time to SHOW WHAT YOU KNOW!! Jot down your choice for each question. After the program, your local teacher will go over the correct answers with you. (Read Show What You Know questions on screen) To get ready to record your observations in the lab, your teacher will give you a different kind of data sheet to copy, with diagrams of the carts, before and after the collisions or explosions. So get ready for thrills, chills, collisions and explosions next time. We ve made a good start today. Let s keep it going and conserve that momentum 9