Chapter 6 Momentum

Momentum Momentum = Mass x Velocity or Momentum = mv Shows the “quantity of motion” of an object. Examples: An elephant has more momentum than a chicken moving at the same speed. A huge ship moving slowly has a huge momentum, and so does a bullet moving fast.

Boulder The boulder has unfortunately for the runner more momentum. Also 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 in mv is zero.

Supertanker Why are the engines of a supertanker normally cut off 25 km from port?

Ship Momentum A supertanker normally cuts off its power when it’s at 25 or so kilometers from port. Because of their huge momentum (due mostly to their huge mass), about 25 kilometers of water resistance are needed to bring it to a halt.

Impulse Changes in momentum may occur when there is either: a change in the mass of an object, a change in velocity, or both. A change in momentum because a velocity change requires acceleration, which in turn requires a net force. The greater the net force that acts on an object, the greater will be the change in velocity and, hence, the change in momentum. The change in momentum is also determined by how long a time the force acts. Apply a force to a cart: briefly produces a small change in its momentum. longer time, a greater change in momentum results. So for changing the momentum of an object, both force and the time during which the force acts are important. The product of force and the time interval the force acts is called impulse.

Impulse Whenever you apply a net force, you also exert an impulse. or in shorthand: Whenever you apply a net force, you also exert an impulse. The resulting change in momentum depends on both the net force and the time interval during which that force acts.

Impulse Actual contact times may be very short.

Impulse-Momentum or In shorthand: We will consider some ordinary examples in which impulse is related to increasing momentum, decreasing momentum over a long time, and decreasing momentum over a short time.

Case 1: Increasing Momentum To increase the momentum of something as much as possible: apply the greatest force you can, 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.

Golf The force that acts on the golf ball in the figure bellow, 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. Impact force against a golf ball.

Impulse When a moving object stops, the impulse would completely remove momentum. This can be accomplished for example by a smaller force acting for a longer time.

Case 2: Decreasing Momentum over a Long Time leads to a Small Force Imagine you are in a car out of control, and you have a choice of slamming into either a concrete wall or a haystack. 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 By hitting the haystack instead of the wall, you extend the time of impact — which is the time during which your momentum is brought to zero. The longer time needs a lesser force to be compensated. If you extend the time of impact 100 times, you reduce the force of impact by 100 times. 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.

Examples: Decreasing Momentum A wrestler thrown down 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 person jumping down bends his knees upon making contact, thereby extending the time during which his momentum is being reduced by 10 to 20 times that of a stiff-legged, abrupt landing. Such knee bending reduces the forces experienced by the bones by 10 to 20 times. Falling on a mat is preferable to falling on the floor, because of the increase of the time of impact. In bungee jumping, the momentum gained during fall must be decreased to zero by an impulse of equal magnitude. The long stretching time of the cord insures a small average force to bring the jumper to a safe halt before hitting the ground. Bungee cords typically stretch to about twice their original length during the fall. Ballet dancers prefer the more elastic wooden floor because it allows a longer time of impact whenever the dancer lands, thus reducing the force of impact. A safety net used by acrobats provides an obvious example of small impact force over a long time to provide the required impulse to reduce the momentum of fall. If you're about to catch a fast ball with your hands, you extend your hands forward so you'll have plenty of room to let your hand move backward after you make contact with the ball. You extend the time of impact and thereby reduce the force of impact. Similarly, a boxer rides or rolls with the punch to reduce the force of impact

Case 3: Decreasing Momentum over a Short Time leads to a Large Force When boxing, and don’t move away from the punch, the “damage” is greater. Likewise if you catch a high speed ball with stiff hands. 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. Catching ball with stiff hands hurts 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 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 vs. Stopping Stopping: 0-(-mv) = +mv Change in momentum = final momentum – initial momentum Stopping: 0-(-mv) = +mv Bouncing: mv-(-mv) = +2mv +

Water Wheel 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. The Pelton wheel. The curved blades cause water to bounce and make a U-turn which produces a greater impulse to turn the wheel.

Example Tennis I = Δp = 1.6 Ns A tennis ball weighing 50 g and moving horizontally with a velocity of 12 m/s, is hit by a racket such that its velocity becomes 20 m/s in exactly the opposite direction. a) What is the change in the momentum of the ball? b) What impulse is delivered by the racket to the ball? I = Δp = 1.6 Ns c) What force is exerted by the racket on the ball if the time of impact is 5 ms? F Δt = Δp therefore F x 0.005 s= =1.6 Ns, F = 320 N

Conservation of Momentum To change the momentum of an object you need a net force. To change the momentum of a system you need a net external force. Note: Internal forces do not count because they appear in equal and opposite pairs that cancel each other out. Examples: molecular forces within a ball have no effect on the momentum of the ball 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. Explanation: these are internal forces, that act and react within the system and cancel each other out. An outside, external force acting on the ball or automobile is required for a change in momentum. If no external force is present, then no change in momentum is possible. Then the total momentum of the system stays constant and is conserved.

Firing A Rifle Pbefore = 0 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. Pafter = Mv-mV=0 Mv = mV

Momentum Features 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. For the system of rifle and bullet, no momentum was gained or was lost. When a physical quantity remains unchanged during a process, that quantity is said to be conserved. Therefore momentum is conserved.

Example Total Momentum A car with mass of 1000kg and velocity of 2 m/s travels eastward. Another car with the mass of 3000kg and velocity of 4 m/s travels westward. What is the total momentum of the cars? v1 = 2m/s v2 = 4m/s Ptot = P1 + P2 Ptot = m1v1 + m2v2 Ptot = 1000kg x 2m/s – 3000kg x 4m/s = = -10000kgm/s

Collisions Total momentum before collision = Total momentum after collision Elastic Collision: no permanent change in shape Inelastic Collision: some permanent change in shape. Completely Inelastic: Bodies stick together after collision.

ConcepTest 6.5a Two Boxes I Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has more momentum after the force acts ? 1) the heavier one 2) the lighter one 3) both the same F light heavy

ConcepTest 6.5a Two Boxes I Two boxes, one heavier than the other, are initially at rest on a horizontal frictionless surface. The same constant force F acts on each one for exactly 1 second. Which box has more momentum after the force acts ? 1) the heavier one 2) the lighter one 3) both the same F light heavy Impulse F Dt = Δp In this case F and Dt are the same for both boxes ! Both boxes will have the same final momentum.

ConcepTest 6.5b Two Boxes II In the previous question, which box has the larger velocity after the force acts? 1) the heavier one 2) the lighter one 3) both the same

ConcepTest 6.5b Two Boxes II In the previous question, which box has the larger velocity after the force acts? 1) the heavier one 2) the lighter one 3) both the same The force is related to the acceleration by Newton’s 2nd Law (F = ma). The lighter box therefore has the greater acceleration and will reach a higher speed after the 1-second time interval. Follow-up: Which box has gone a larger distance after the force acts? Follow-up: Which box has gained more KE after the force acts?

ConcepTest 6.6 Watch Out! You drive around a curve in a narrow one-way street at 30 mph when you see an identical car heading straight toward you at 30 mph. You have two options: hit the car head-on or swerve into a massive concrete wall (also head-on). What should you do? 1) hit the other car 2) hit the wall 3) makes no difference 4) call your physics teacher!! 5) get insurance!

ConcepTest 6.6 Watch Out! You drive around a curve in a narrow one-way street at 30 km/h when you see an identical car heading straight toward you at 30 km/h. You have two options: hit the car head-on or swerve into a massive concrete wall (also head-on). What should you do? 1) hit the other car 2) hit the wall 3) makes no difference 4) call your physics teacher!! 5) get insurance! In both cases your momentum will decrease to zero in the collision. Given that the time Dt of the collision is the same, then the force exerted on YOU will be the same!! If a truck is approaching at 30 km/h, then you’d be better off hitting the wall in that case. On the other hand, if it’s only a mosquito, well, you’d be better off running him down...

ConcepTest 6.7 Impulse A small beanbag and a bouncy rubber ball are dropped from the same height above the floor. They both have the same mass. Which one will impart the greater impulse to the floor when it hits? 1) the beanbag 2) the rubber ball 3) both the same

ConcepTest 6.7 Impulse A small beanbag and a bouncy rubber ball are dropped from the same height above the floor. They both have the same mass. Which one will impart the greater impulse to the floor when it hits? 1) the beanbag 2) the rubber ball 3) both the same Both objects reach the same speed at the floor. However, while the beanbag comes to rest on the floor, the ball bounces back up with nearly the same speed as it hit. Thus, the change in momentum for the ball is greater, because of the rebound. The impulse delivered by the ball is twice that of the beanbag. For the beanbag: Dp = pf – pi = 0 – (–mv ) = mv For the rubber ball: Dp = pf – pi = mv – (–mv ) = 2mv Follow-up: Which one imparts the larger force to the floor?

ConcepTest 6.8 Singing in the Rain A person stands under an umbrella during a rainstorm. Later the rain turns to hail, although the number of “drops” hitting the umbrella per time and their speed remains the same. Which case requires more force to hold the umbrella? 1) when it is hailing 2) when it is raining 3) same in both cases

ConcepTest 6.8 Singing in the Rain A person stands under an umbrella during a rainstorm. Later the rain turns to hail, although the number of “drops” hitting the umbrella per time and their speed remains the same. Which case requires more force to hold the umbrella? 1) when it is hailing 2) when it is raining 3) same in both cases When the raindrops hit the umbrella, they tend to splatter and run off, whereas the hailstones hit the umbrella and bounce back upward. Thus, the change in momentum (impulse) is greater for the hail. Since Dp = F Dt, more force is required in the hailstorm. This is similar to the situation with the bouncy rubber ball in the previous question.

ConcepTest 6.9a Going Bowling I A bowling ball and a ping-pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest? 1) the bowling ball 2) same time for both 3) the ping-pong ball 4) impossible to say p

ConcepTest 6.9a Going Bowling I A bowling ball and a ping-pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest? 1) the bowling ball 2) same time for both 3) the ping-pong ball 4) impossible to say p since Dp = F Dt and F and Dp are the same for both balls! It will take the same amount of time to stop them.

ConcepTest 6.9b Going Bowling II A bowling ball and a ping-pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? 1) the bowling ball 2) same distance for both 3) the ping-pong ball 4) impossible to say p

ConcepTest 6.9b Going Bowling II A bowling ball and a ping-pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater? 1) the bowling ball 2) same distance for both 3) the ping-pong ball 4) impossible to say Use the work-energy theorem: W = DKE. The ball with less mass has the greater speed (why?), and thus the greater KE (why again?). In order to remove that KE, work must be done, where W = Fd. Since the force is the same in both cases, the distance needed to stop the less massive ball must be bigger. p

Inelastic Collision By simple algebra, V = 5 m/s. Inelastic collision. The momentum of the freight car on the left is shared with the freight car on the right after collision. 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.

Sign of Momentum 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. In the inelastic collisions shown in figure bellow, 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. If, on the other hand, A and B are moving in the same direction (A catching up with B), the total momentum is simply the addition of their individual momenta (Ex. 10+0=10).

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?

Fish Suppose the small fish 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:

Elastic Collision Elastic collisions of equally massive balls. (a) A green ball strikes a yellow ball at rest. (b) A head-on collision. (c) A collision of balls moving in the same direction. In each case, momentum is transferred from one ball to the other.

Example Elastic Collision A 0.7 kg cart moving with 2m/s collides elastically with a 0.5 kg cart at rest. If the velocity of the first cart after the collision is 0.4 m/s, find the velocity of the second car immediately after the collision? Before After (Total Momentum)before = (Total Momentum)after p’1 + p’2 p1 + 0 = 0.7kg x 2m/s 0.7kg x 0.4m/s +0.5kg x v’2 = 1.4 m/s = 0.28 m/s + 0.5v’2 v’2 = 2.24 m/s

More Complicated Collisions The total momentum remains unchanged in any collision, regardless of the angle between the tracks of the colliding objects. Expressing the total momentum when different directions are involved can be achieved with the parallelogram rule of vector addition. Momentum is a vector quantity.

Explosion The figure bellow shows a falling firecracker exploding into two pieces. The momenta of the fragments combine by vector addition to equal the original momentum of the falling firecracker. Explosion: Reversed completely inelastic collision

Summary of Terms Momentum The product of the mass of an object and its velocity.  Impulse The product of the force acting on an object and the time during which it acts. In an interaction, impulses are equal and opposite.  Relationship of impulse and momentum Impulse is equal to the change in the momentum of the object that the impulse acts on. In symbol notation,   Conservation of momentum When no external net force acts on an object or a system of objects, no change of momentum takes place. Hence, the momentum before an event involving only internal forces is equal to the momentum after the event:  Elastic collision A collision in which colliding objects rebound without lasting deformation or the generation of heat.  Inelastic collision A collision in which the colliding objects change shape, generate heat, and possibly stick together.

Whenever an interaction occurs in a system, forces occur in equal and opposite pairs. Which of the following do not always occur in equal and opposite pairs? 1. Impulses. 2. Accelerations. 3. Momentum changes. 4. All of these occur in equal and opposite pairs. 5. None of these do. Ch 6-1

H Momentum Due

Whenever an interaction occurs in a system, forces occur in equal and opposite pairs. Which of the following do not always occur in equal and opposite pairs? 1. Impulses. 2. Accelerations. 3. Momentum changes. 4. All of these occur in equal and opposite pairs. 5. None of these do. Ch 6-1 Answer: 2 Because time for each interaction part is the same, impulses and momentum changes also occur in equal and opposite pairs. But not necessarily accelerations, because the masses of the interaction may differ. Consider equal and opposite forces acting on masses of different magnitude.

Which would be more damaging? 1. Driving into a massive concrete wall. 2. Driving at the same speed into a head- on collision with an identical car traveling toward you at the same speed. 3. They are equivalent. Ch 6-2

Which would be more damaging? 1. Driving into a massive concrete wall. 2. Driving at the same speed into a head- on collision with an identical car traveling toward you at the same speed. 3. They are equivalent. Ch 6-2 Answer: 3 Your car decelerates to a dead stop either way. The dead stop is easy to see when hitting the wall, and a little thought will show the same is true when hitting the car. If the oncoming car were traveling more slowly, with less momentum, you’d keep going after the collision with more “give,” and less damage (to you). But if the oncoming car had more momentum than you, it would keep going and you’d snap into a sudden reverse with greater damage. Identical cars at equal speeds means equal momenta—zero before, zero after collision.

Strictly speaking, when a gun is fired, compared with the momentum of the recoiling gun, the opposite momentum of the bullet is 1. less. 2. more. 3. the same. Ch 6-4 Thanks to David G. Willey.

Strictly speaking, when a gun is fired, compared with the momentum of the recoiling gun, the opposite momentum of the bullet is 1. less. 2. more. 3. the same. Ch 6-4 Thanks to David G. Willey Answer: 1 Why? Because more than just a bullet comes out of the barrel when a gun is fired. The gas, formed when the powder in the cartridge burns, pushes the bullet along the barrel, and this gas too has appreciable mass and exits at high speed. More than negligible momentum is given to the gases. So, momentum of recoiling gun = momentum of bullet + momentum of gases More than one person has been accidentally killed by a “blank” fired at close range!

1. to the left (backward). 2. to the right (forward). 3. not at all. An ice sailcraft is stalled on a frozen lake on a windless day. A large fan blows air into the sail. If the wind produced by the fan strikes and bounces backward from the sail, the sailcraft will move Ch 6-5 1. to the left (backward). 2. to the right (forward). 3. not at all.

1. to the left (backward). 2. to the right (forward). 3. not at all. An ice sailcraft is stalled on a frozen lake on a windless day. A large fan blows air into the sail. If the wind produced by the fan strikes and bounces backward from the sail, the sailcraft will move Ch 6-5 Answer: 2 You might think the sailcraft wouldn't move— that the force of wind impact on the sail would be balanced by the reaction force on the fan— which would be true if the wind came to an abrupt halt upon striking the sail. But it doesn’t. The wind bounces from the sail and produces a greater force on the sail than if it merely stopped (like any collision, more force is required to reverse the direction of something than to merely start or stop it). So there is a net force on the sailcraft and a forward acceleration. Or consider impulse and momentum. The impulse on the sail is greater than the impulse on the fan. Why? Because the air undergoes more change in momentum bouncing from the sail than starting from the fan. Note there are two force pairs to consider: (1) the fan-air force pair, and (2) the air-sail force pair. Because of bouncing, the air-sail pair is greater. Solid vectors show forces exerted on the sailcraft; dashed vectors show forces exerted on the air. The net force on the sailcraft is forward, to the right. 1. to the left (backward). 2. to the right (forward). 3. not at all.

Which has a greater momentum, a heavy truck at rest or a moving skateboard? Same

Which has a greater momentum, a heavy truck at rest or a moving skateboard? Same

Which undergoes the greatest change in momentum: a baseball that is caught, a baseball that is thrown, or a baseball that is caught and then thrown back, if the baseballs have the same speed just before being caught and just after being thrown?

Which undergoes the greatest change in momentum: a baseball that is caught, a baseball that is thrown, or a baseball that is caught and then thrown back, if the baseballs have the same speed just before being caught and just after being thrown?

Which undergoes the greatest change in impulse: a baseball that is caught, a baseball that is thrown, or a baseball that is caught and then thrown back, if the baseballs have the same speed just before being caught and just after being thrown?

Which undergoes the greatest change in impulse: a baseball that is caught, a baseball that is thrown, or a baseball that is caught and then thrown back, if the baseballs have the same speed just before being caught and just after being thrown?

Would momentum be conserved for the system of rifle and bullet if momentum were not a vector quantity? Yes No I don’t know

Would momentum be conserved for the system of rifle and bullet if momentum were not a vector quantity? Yes No! I don’t know

For which type of collision is momentum conserved? elastic collision inelastic collision both

For which type of collision is momentum conserved? elastic collision inelastic collision both

Railroad car A rolls at a certain speed and makes a perfectly elastic collision with car B of the same mass. After the collision, car A is observed to be at rest. How does the speed of car B compare with the initial speed of car A? Greater than Less than Same Zero

Railroad car A rolls at a certain speed and makes a perfectly elastic collision with car B of the same mass. After the collision, car A is observed to be at rest. How does the speed of car B compare with the initial speed of car A? Greater than Less than Same Zero

If the equally massive cars of the previous question stick together after colliding inelastically, how does their speed after the collision compare with the initial speed of car A? Double Half Same Zero

If the equally massive cars of the previous question stick together after colliding inelastically, how does their speed after the collision compare with the initial speed of car A? Double Half Same Zero

Suppose a ball of putty moving horizontally with 1 kg m/s of momentum collides and sticks to an identical ball of putty moving vertically with 1 kg m/s of momentum. What is the total momentum of the balls of putty in kg m/s before and after the collision? 2 √2

Suppose a ball of putty moving horizontally with 1 kg m/s of momentum collides and sticks to an identical ball of putty moving vertically with 1 kg m/s of momentum. What is the total momentum of the balls of putty in kg m/s before and after the collision? 2 √2 √2 1 1