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Momentum A heavy truck is harder to stop than a small car moving at the same speed. We state this fact by saying that the truck has more momentum than the car. By momentum we mean inertia in motion. More specifically momentum is defined as the product of the mass of an object and its velocity. Which we abbreviate mv. Moving Object A moving object can have a large momentum if either its mass or its velocity is large, or if both its mass and its velocity are large. The truck has more momentum than the car moving at the same speed because it has a greater mass. We can see that a huge ship moving at a small speed can have a large momentum, as can a small bullet moving at a high speed. And, of course, a huge object moving at a high speed, such as a massive truck rolling down a steep hill with no brakes, has a huge momentum, whereas the same truck at rest has no momentum at all—because the v part of mv is zero. Why are the engines of a supertanker normally cut off 25 km from port? Force and Momentum Changes in momentum may occur when there is either a change in the mass of an object, a change in velocity, or both. If momentum changes while the mass remains unchanged, as is most often the case, then the velocity changes. Acceleration occurs. And what produces an acceleration? The answer is a force. The greater the force that acts on an object, the greater will be the change in velocity and, hence, the change in momentum. Time and Momentum But something else is important in changing momentum: time—how long a time the force acts. Apply a force briefly to a stalled automobile, and you produce a small change in its momentum. Apply the same force over an extended period of time, and a greater change in momentum results. A long sustained force produces more change in momentum than the same force applied briefly. Force and Time-Impulse So for changing the momentum of an object, both force and the time during which the force acts are important. We name the product of force and this time interval impulse. Whenever you exert a net force on something, you also exert an impulse. The resulting acceleration depends on the net force; the resulting change in momentum depends on both the net force and the time during which that force acts. Check Yourself 1. Which has more momentum, a 1-ton car moving at 100 km/h or a 2-ton truck moving at 50 km/h? 2. Does a moving object have impulse? 3. Does a moving object have momentum? Impulse Changes Momentum Impulse changes momentum in much the same way that force changes velocity. The relationship of impulse to change of momentum comes from Newton's second law (a = F/m). The time interval of impulse is “buried” in the term for acceleration (change in velocity/time interval). Rearrangement of Newton's second law gives “force multiplied by the time during which it acts equals change in momentum.” Impulse-Momentum The impulse-momentum relationship helps us to analyze many examples in which forces act and motion changes. Sometimes the impulse can be considered to be the cause of a change of momentum. Sometimes a change of momentum can be considered to be the cause of an impulse. It doesn't matter which way you think about it. The important thing is that impulse and change of momentum are always linked. Here we will consider some ordinary examples in which impulse is related to - Case 1-increasing momentum If you wish to increase the momentum of something as much as possible, you not only apply the greatest force you can, you also extend the time of application as much as possible. Long-range cannons have long barrels. The longer the barrel, the greater the velocity of the emerging cannonball or shell. Why? The force of exploding gunpowder in a long barrel acts on the cannonball for a longer time. This increased impulse produces a greater momentum. Of course the force that acts on the cannonball is not steady—it is strong at first and weaker as the gases expand. In this and many other cases the forces involved in impulses vary over time. The force that acts on the golf ball, for example, increases rapidly as the ball is distorted and then diminishes as the ball comes up to speed and returns to its original shape. When we speak of impact forces in this chapter, we mean the average force of impact. Case 2-Decreasing Momentum Over a Long Time Imagine you are in a car out of control, and you have a choice of slamming into either a concrete wall or a haystack. You needn't know much physics to make the better decision, but knowing some physics helps you to understand why hitting something soft is entirely different from hitting something hard. In the case of hitting either the wall or the haystack, your momentum will be decreased by the same amount, and this means that the impulse needed to stop you is the same. The same impulse means the same product of force and time—not the same force or the same time. You have a choice. By hitting the haystack instead of the wall, you extend the time of impact—you extend the time during which your momentum is brought to zero. The longer time is compensated by a lesser force. If you extend the time of impact 100 times, you reduce the force of impact by 100. So whenever you wish the force of impact to be small, extend the time of impact. A large change in momentum in a long time requires a small force. A wrestler thrown to the floor tries to extend his time of arrival on the floor by relaxing his muscles and spreading the crash into a series of impacts as foot, knee, hip, ribs, and shoulder fold onto the floor in turn. The increased time of impact reduces the force of impact. A large change in momentum in a short time requires a large force. The boxer's jaw provides an impulse that reduces the momentum of the punch. (a) The boxer is moving away when the glove hits, thereby extending the time of contact. This means the force is less than if the boxer had not moved. (b) The boxer is moving into the glove, thereby lessening the time of contact. This means that the force is greater than if the boxer had not moved. Case 3- Decreasing Momentum Over a Short Time When boxing, move into a punch instead of away, and you're in trouble. Likewise if you catch a high-speed baseball while your hand moves toward the ball instead of away upon contact. Or when out of control in a car, drive it into a concrete wall instead of a haystack and you're really in trouble. In these cases of short impact times, the impact forces are large. Remember that for an object brought to rest, the impulse is the same, no matter how it is stopped. But if the time is short, the force will be large. The idea of short time of contact explains how a karate expert can sever a stack of bricks with the blow of her bare hand . She brings her arm and hand swiftly against the bricks with considerable momentum. This momentum is quickly reduced when she delivers an impulse to the bricks. The impulse is the force of her hand against the bricks multiplied by the time her hand makes contact with the bricks. By swift execution she makes the time of contact very brief and correspondingly makes the force of impact huge. If her hand is made to bounce upon impact, the force is even greater. Check Yourself 1. If the boxer is able to make the duration of impact three times as long by riding with the punch, by how much will the force of impact be reduced? 2. If the boxer instead moves into the punch such as to decrease the duration of impact by half, by how much will the force of impact be increased? 3. A boxer being hit with a punch contrives to extend time for best results, whereas a karate expert delivers a force in a short time for best results. Isn't there a contradiction here? 4. When does impulse equal momentum? Bouncing You know that if a flower pot falls from a shelf onto your head, you may be in trouble. And whether you know it or not, if it bounces from your head, you may be in more serious trouble. Impulses are greater when bouncing takes place. This is because the impulse required to bring something to a stop and then, in effect, “throw it back again” is greater than the impulse required merely to bring something to a stop. Suppose, for example, that you catch the falling pot with your hands. Then you provide an impulse to catch it and reduce its momentum to zero. If you were to then throw the pot upward, you would have to provide additional impulse. So it would take more impulse to catch it and throw it back up than merely to catch it. The same greater impulse is supplied by your head if the pot bounces from it. The Pelton Wheel An interesting application of the greater impulse that occurs when bouncing takes place was employed with great success in California during the gold rush days. The water wheels used in gold-mining operations were ineffective. A man named Lester A. Pelton saw that the problem had to do with their flat paddles. He designed curved-shape paddles that would cause the incident water to make a U-turn upon impact—to “bounce.” In this way the impulse exerted on the water wheels was greatly increased. Pelton patented his idea and made more money from his invention, the Pelton wheel, than most gold miners made from gold. Check Yourself Check Yourself 1. How does the force that the woman exerts on the bricks compare with the force exerted on her hand? 2. How will the impulse resulting from the impact differ if her hand bounces back upon striking the bricks? External Impulse Changes Momentum Recall what Newton's second law tells us: If we want to accelerate an object, we must apply a force. We say much the same thing in this chapter when we say that to change the momentum of an object we must apply an impulse. In any case, the force or impulse must be exerted on the object or any system of objects by something external to the object or system. Internal forces don't count. Internal Forces do not Change Momentum For example, the molecular forces within a baseball have no effect on the momentum of the baseball, just as a person sitting inside an automobile pushing against the dashboard, with the dashboard pushing back, has no effect in changing the momentum of the automobile. This is because these forces are internal forces, that is, forces that act and react within the systems themselves. An outside, or external, force acting on the baseball or automobile is required for a change in momentum. If no external force is present, then no change in momentum is possible. Conservation of Momentum When a bullet is fired from a rifle, the forces present are internal forces. The total momentum of the system comprising the bullet and rifle therefore undergoes no net change. By Newton's third law of action and reaction, the force exerted on the bullet is equal and opposite to the force exerted on the rifle. The forces acting on the bullet and rifle act for the same time, resulting in equal but oppositely directed impulses and therefore equal and oppositely directed momenta (the plural form of momentum). Conservation of Momentum The recoiling rifle has just as much momentum as the speeding bullet. Although both the bullet and rifle by themselves have gained considerable momentum, the bullet and rifle together as a system experience no net change in momentum. Before firing, the momentum was zero; after firing, the net momentum is still zero. No momentum is gained and no momentum is lost. The momentum before firing is zero. After firing, the net momentum is still zero, because the momentum of the rifle is equal and opposite to the momentum of the bullet. Rifle-Bullet Example Two important ideas are to be learned from the rifle-and-bullet example. The first is that momentum, like velocity, is a quantity that is described by both magnitude and direction; we measure both “how much” and “which direction.” Momentum is a vector quantity. Therefore, when momenta act in the same direction, they are simply added; when they act in opposite directions, they are subtracted. Rifle-Bullet Example The second important idea to be taken from the rifle-and-bullet example is the idea of conservation. The momentum before and after firing is the same. For the system of rifle and bullet, no momentum was gained; none was lost. When a physical quantity remains unchanged during a process, that quantity is said to be conserved. We say momentum is conserved. Check Yourself 1. A high-speed bus and an innocent bug have a head-on collision. The sudden change of momentum for the bug spatters it all over the windshield. Is the change in momentum of the bus greater, less, or the same as the change in momentum experienced by the unfortunate bug? 2. The Starship Enterprise is being pursued in space by a Klingon warship moving at the same speed. What happens to the speed of the Klingon ship when it fires a projectile at the Enterprise? What happens to the speed of the Enterprise when it returns the fire? Collisions Momentum is conserved in collisions—that is, the net momentum of a system of colliding objects is unchanged before, during, and after the collision. This is because the forces that act during the collision are internal forces—forces acting and reacting within the system itself. There is only a redistribution or sharing of whatever momentum exists before the collision. In any collision, we can say Net momentum before collision = net momentum after collision. This is true no matter how the objects might be moving before they collide. Elastic Collision When a moving billiard ball makes a headon collision with another billiard ball at rest, the moving ball comes to rest and the other ball moves with the speed of the colliding ball. We call this an elastic collision; ideally the colliding objects rebound without lasting deformation or the generation of heat. Inelastic Collision But momentum is conserved even when the colliding objects become entangled during the collision. This is an inelastic collision, characterized by deformation or the generation of heat or both. In a perfectly inelastic collision, both objects stick together. Consider, for example, the case of a freight car moving along a track and colliding with another freight car at rest. If the freight cars are of equal mass and are coupled by the collision, can we predict the velocity of the coupled cars after impact? Inelastic Collision Inelastic collision. The momentum of the freight car on the left is shared with the freight car on the right after collision. Conservation of Momentum Suppose the single car is moving at 10 meters per second, and we consider the mass of each car to be m. Then, from the conservation of momentum, By simple algebra, V = 5 m/s. This makes sense, for since twice as much mass is moving after the collision, the velocity must be half as much as the velocity before collision. Both sides of the equation are then equal. Direction is important Notice the importance of direction in these cases. As with any pair of vectors, momenta in the same direction are simply added. If two objects are moving toward each other, one of the momenta is considered negative, and we combine them by subtraction. Inelastic Collision Inelastic collisions. The net momentum of the trucks before and after collision is the same. The same principle applies to gently docking spacecraft. Their net momentum just before docking is preserved as their net momentum just after docking. Check Yourself Consider an air track. Suppose a gliding cart with a mass of 0.5 kg bumps into and sticks to a stationary cart that has a mass of 1.5 kg. If the speed of the gliding cart before impact is v before, how fast will the coupled carts glide after collision? Examples For a numerical example of momentum conservation, consider a fish that swims toward and swallows a smaller fish at rest. If the larger fish has a mass of 5 kg and swims 1 m/s toward a 1-kg fish, what is the velocity of the larger fish immediately after lunch? We will neglect the effects of water resistance. Conservation of Momentum Two fish make up a system, which has the same momentum just before lunch and just after lunch. Complicated Collisions The net momentum remains unchanged in any collision, regardless of the angle between the tracks of the colliding objects. Expressing the net momentum when different directions are involved can be achieved with the parallelogram rule of vector addition. We will show some simple examples to convey the concept. Momentum is a vector quantity After the firecracker bursts, the momenta of its fragments add up (by vector addition) to the original momentum. Whatever the nature of a collision or however complicated it is, the total momentum before, during, and after remains unchanged. This extremely useful law enables us to learn much from collisions without knowing any details about the interaction forces that act in the collision. We will see in the next chapter that energy as well as momentum is conserved. By applying momentum and energy conservation to the collisions of subatomic particles as observed in various detection chambers, we can compute the masses of these tiny particles. We obtain this information by measuring momenta and energy before and after collisions. Remarkably, this achievement is possible without any exact knowledge of the forces that act. Conservation of momentum and conservation of energy (next chapter) are the two most powerful tools of mechanics. Applying them yields detailed information that ranges from facts about the interactions of subatomic particles to the structure and motion of entire galaxies. animations Momentum Quiz 4 1. a. What is the momentum of an 8 Kg bowling ball rolling at 3 m / s ? b. If the bowling ball rolls into a pillow and stops in 0.2 s, calculate the average force it exerts on the pillow. 2. a. What impulse occurs when an average force of 15 N is exerted on a cart for 3 s? b. If the mass of the cart is 3 Kg and the cart is initially at rest calculate its final speed. Quiz 5 1. A 40 Kg projectile leaves a 2000 Kg Launcher with a speed of 300m/s. What is the recoil speed of the launcher? 2. An 9000Kg truck moving at 10 m / s collides with a 1500 Kg car moving at 20 m / s in the opposite direction. If they stick together after impact, how fast, and in what direction, will they be moving? Momentum A heavy truck is harder to stop than a small car moving at the same speed. We state this fact by saying that the truck has more momentum than the car. By momentum - Force and Momentum Changes in momentum may occur when there is either a change in the mass of an object, a change in velocity, or both. If momentum changes while the mass remains unchanged, as is most often the case, then the velocity changes. Acceleration occurs. And what produces an acceleration? The answer is a force. The greater the force that acts on an object, the greater will be the change in velocity and, hence, the change in momentum. Time and Momentum But something else is important in changing momentum: time—how long a time the force acts. Apply a force briefly to a stalled automobile, and you produce a small change in its momentum. Apply the same force over an extended period of time, and a greater change in momentum results. A long sustained force produces more change in momentum than the same force applied briefly. Force and Time-Impulse So for changing the momentum of an object, both force and the time during which the force acts are important. We name the product of force and this time interval impulse. Whenever you exert a net force on something, you also exert an impulse. The resulting acceleration depends on the net force; the resulting change in momentum depends on both the net force and the time during which that force acts. Check Yourself 1. Which has more momentum, a 1-ton car moving at 100 km/h or a 2-ton truck moving at 50 km/h? 2. Does a moving object have impulse? 3. Does a moving object have momentum? Impulse Changes Momentum Impulse changes momentum in much the same way that force changes velocity. The relationship of impulse to change of momentum comes from Newton's second law (a = F/m). The time interval of impulse is “buried” in the term for acceleration (change in velocity/time interval). Rearrangement of Newton's second law gives “force multiplied by the time during which it acts equals change in momentum.” Truck into Haystack A large change in momentum in a long time requires a small force. Truck into Wall A large change in momentum in a short time requires a large force. Check Yourself 1. If the boxer is able to make the duration of impact three times as long by riding with the punch, by how much will the force of impact be reduced? 2. If the boxer instead moves into the punch such as to decrease the duration of impact by half, by how much will the force of impact be increased? 3. A boxer being hit with a punch contrives to extend time for best results, whereas a karate expert delivers a force in a short time for best results. Isn't there a contradiction here? 4. When does impulse equal momentum? Bouncing You know that if a flower pot falls from a shelf onto your head, you may be in trouble. And whether you know it or not, if it bounces from your head, you may be in more serious trouble. Impulses are greater when bouncing takes place. This is because the impulse required to bring something to a stop and then, in effect, “throw it back again” is greater than the impulse required merely to bring something to a stop. Suppose, for example, that you catch the falling pot with your hands. Then you provide an impulse to catch it and reduce its momentum to zero. If you were to then throw the pot upward, you would have to provide additional impulse. So it would take more impulse to catch it and throw it back up than merely to catch it. The same greater impulse is supplied by your head if the pot bounces from it. Check Yourself Check Yourself 1. How does the force that the woman exerts on the bricks compare with the force exerted on her hand? 2. How will the impulse resulting from the impact differ if her hand bounces back upon striking the bricks? Karate Chop External Impulse Changes Momentum Recall what Newton's second law tells us: If we want to accelerate an object, we must apply a force. We say much the same thing in this chapter when we say that to change the momentum of an object we must apply an impulse. In any case, the force or impulse must be exerted on the object or any system of objects by something external to the object or system. Internal forces don't count. Internal Forces do not Change Momentum For example, the molecular forces within a baseball have no effect on the momentum of the baseball, just as a person sitting inside an automobile pushing against the dashboard, with the dashboard pushing back, has no effect in changing the momentum of the automobile. This is because these forces are internal forces, that is, forces that act and react within the systems themselves. An outside, or external, force acting on the baseball or automobile is required for a change in momentum. If no external force is present, then no change in momentum is possible. Conservation of Momentum When a bullet is fired from a rifle, the forces present are internal forces. The total momentum of the system comprising the bullet and rifle therefore undergoes no net change. By Newton's third law of action and reaction, the force exerted on the bullet is equal and opposite to the force exerted on the rifle. The forces acting on the bullet and rifle act for the same time, resulting in equal but oppositely directed impulses and therefore equal and oppositely directed momenta (the plural form of momentum). Conservation of Momentum The recoiling rifle has just as much momentum as the speeding bullet. Although both the bullet and rifle by themselves have gained considerable momentum, the bullet and rifle together as a system experience no net change in momentum. Before firing, the momentum was zero; after firing, the net momentum is still zero. No momentum is gained and no momentum is lost Rifle-Bullet Example Two important ideas are to be learned from the rifle-and-bullet example. The first is that momentum, like velocity, is a quantity that is described by both magnitude and direction; we measure both “how much” and “which direction.” Momentum is a vector quantity. Therefore, when momenta act in the same direction, they are simply added; when they act in opposite directions, they are subtracted Rifle-Bullet Example The second important idea to be taken from the rifle-and-bullet example is the idea of conservation. The momentum before and after firing is the same. For the system of rifle and bullet, no momentum was gained; none was lost. When a physical quantity remains unchanged during a process, that quantity is said to be conserved. We say momentum is conserved. Check Yourself 1. A high-speed bus and an innocent bug have a head-on collision. The sudden change of momentum for the bug spatters it all over the windshield. Is the change in momentum of the bus greater, less, or the same as the change in momentum experienced by the unfortunate bug? 2. The Starship Enterprise is being pursued in space by a Klingon warship moving at the same speed. What happens to the speed of the Klingon ship when it fires a projectile at the Enterprise? What happens to the speed of the Enterprise when it returns the fire? Collisions Momentum is conserved in collisions—that is, the net momentum of a system of colliding objects is unchanged before, during, and after the collision. This is because the forces that act during the collision are internal forces—forces acting and reacting within the system itself. There is only a redistribution or sharing of whatever momentum exists before the collision. In any collision, we can say Net momentum before collision = net momentum after collision. This is true no matter how the objects might be moving before they collide. Elastic Collisions When a moving billiard ball makes a headon collision with another billiard ball at rest, the moving ball comes to rest and the other ball moves with the speed of the colliding ball. We call this an elastic collision; ideally the colliding objects rebound without lasting deformation or the generation of heat. Inelastic Collision But momentum is conserved even when the colliding objects become entangled during the collision. This is an inelastic collision, characterized by deformation or the generation of heat or both. In a perfectly inelastic collision, both objects stick together. Consider, for example, the case of a freight car moving along a track and colliding with another freight car at rest. If the freight cars are of equal mass and are coupled by the collision, can we predict the velocity of the coupled cars after impact? Check Yourself Consider an air track. Suppose a gliding cart with a mass of 0.5 kg bumps into and sticks to a stationary cart that has a mass of 1.5 kg. If the speed of the gliding cart before impact is v before, how fast will the coupled carts glide after collision? Example For a numerical example of momentum conservation, consider a fish that swims toward and swallows a smaller fish at rest. If the larger fish has a mass of 5 kg and swims 1 m/s toward a 1-kg fish, what is the velocity of the larger fish immediately after lunch? We will neglect the effects of water resistance. Direction is Important As with any pair of vectors, momenta in the same direction are simply added. If two objects are moving toward each other, one of the momenta is considered negative, and we combine them by subtraction. Complicated Collisions The net momentum remains unchanged in any collision, regardless of the angle between the tracks of the colliding objects. Expressing the net momentum when different directions are involved can be achieved with the parallelogram rule of vector addition. We will not treat such complicated cases in great detail here but will show some simple examples to convey the concept Whatever the nature of a collision or however complicated it is, the total momentum before, during, and after remains unchanged. This extremely useful law enables us to learn much from collisions without knowing any details about the interaction forces that act in the collision. We will see in the next chapter that energy as well as momentum is conserved. By applying momentum and energy conservation to the collisions of subatomic particles as observed in various detection chambers, we can compute the masses of these tiny particles. We obtain this information by measuring momenta and energy before and after collisions. Remarkably, this achievement is possible without any exact knowledge of the forces that act. Conservation of momentum and conservation of energy (next chapter) are the two most powerful tools of mechanics. Applying them yields detailed information that ranges from facts about the interactions of subatomic particles to the structure and motion of entire galaxies.