Impulse & Momentum

Impulse & Momentum

? Both the truck and the bicycle lost their brakes on a downhill slope and rolling slowly downwards, and the drivers are shouting “help! help!”. Would you be able to stop the truck? What it has more compared to the bicycle that you would not be able to stop it?

? Both the truck and the bicycle lost their brakes on a downhill slope and rolling slowly downwards, and the drivers are shouting “help! help!”. Would you be able to stop the truck? What it has more compared to the bicycle that you would not be able to stop it? Inertia

Momentum momentum is the inertia in motion. More specifically momentum is defined as the product of the mass of an object and its velocity; that is,

Momentum We all know that 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.

Momentum 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.

Change in momentum related to force 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 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.

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. Or, in shorthand notation, 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.

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 and a = F/m the definition of accelaration a = Δv/t F/m = Δv/t Ft = m Δv Ft = Δmv which reads, “ force multiplied by the time during which it acts equals change in momentum”.

Impulse Changes 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 (1) increasing momentum, (2) decreasing momentum over a long time, and (3) decreasing momentum over a short time.

Momentum • Which has more momentum, a 1-ton car moving at 100 km/h or a 2-ton truck moving at 50 km/h?

Momentum • Both have the same momentum.

Impulse 2. Does a moving object have impulse?

Impulse 2. No, impulse is not something an object has, like momentum. Impulse is what an object can provide or what it can experience when it interacts with some other object. An object cannot possess impulse just as it cannot possess force.

Momentum 3. Does a moving object have momentum?

Momentum 3. Yes, but, like velocity, in a relative sense—that is, with respect to a frame of reference, often taken to be the Earth's surface. The momentum possessed by a moving object with respect to a stationary point on Earth may be quite different from the momentum it possesses with respect to another moving object.

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. Hence the different results in pushing briefly on a stalled automobile and giving it a sustained push.

Increasing Momentum Long-range cannons have long barrels. The longer the barrel, the greater the velocity of the emerging cannonball or shell. The force of exploding gunpowder in a long barrel acts on the cannonball for a longer time. This increased impulse produces a greater momentum.

Increasing Momentum 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.

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.

Decreasing Momentum over a Long Time 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.

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.

Decreasing Momentum over a Short Time 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.

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.

Bouncing 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.

When does impulse equal momentum? Impulse equals a change in momentum. if an object is brought to rest, impulse = initial momentum. If the initial momentum of an object is zero when the impulse is applied, then impulse = final momentum. Total impulse = initial momentum + final momentum.

Conservation of Momentum 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. 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). 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.

Conservation of Momentum 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). 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.

Vector 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. 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.

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 head-on 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?

conservation of momentum Suppose the single car is moving at 10 m/s, 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.

inelastic collision The net momentum of the trucks before and after collision is the same. Note the inelastic collisions shown in Figure. If A and B are moving with equal momenta in opposite directions (A and B colliding head-on), then one of these is considered to be negative, and the momenta add algebraically to zero. After collision, the coupled wreck remains at the point of impact, with zero momentum.

inelastic collision If, on the other hand, A and B are moving in the same direction (A catching up with B), the net momentum is simply the addition of their individual momenta.

momentum conservation 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 1kg fish, what is the velocity of the larger fish immediately after lunch? We will neglect the effects of water resistance.

momentum conservation

momentum conservation Suppose the small fish in the preceding example is not at rest, but swims toward the left at a velocity of 4 m/s. It swims in a direction opposite that of the larger fish—a negative direction, if the direction of the larger fish is considered positive. In this case,

momentum conservation Suppose the small fish in the preceding example is not at rest, but swims toward the left at a velocity of 8 m/s. It swims in a direction opposite that of the larger fish—a negative direction, if the direction of the larger fish is considered positive. In this case, The final velocity is −1/2 m/s. What is the significance of the minus sign? It means that the final velocity is opposite to the initial velocity of the larger fish.

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. Car A has a momentum directed due east, and car B's momentum is directed due north. If their individual momenta are equal in magnitude, then their combined momentum is in a northeast direction. Diagonal is √2 times the length of the side of a square.

Component momenta A firecracker exploding into two pieces. The momenta of the fragments combine by vector addition to equal the original momentum of the falling firecracker.

Learn from Collisions 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.

Impulse & Momentum