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AP Physics – Conservation of Momentum We’ve studied the law of conservation of energy and found it to be a very powerful thing in the world of physics. Another very powerful conservation law has to do with momentum. This is the law of conservation of momentum. Momentum is Conserved in an isolated system. “Isolated system” means that there are no external forces acting on the thing. The type of interaction that involves changes to momentum that we will deal with are called collisions. (Although not all of them are what we think of as a collision, as we shall see.) The law of conservation of momentum means that for a collision in an isolated system, momentum must be conserved. This means that the total momentum of the system before the collision must be equal to the total momentum after the collision. p p' Here p is the momentum before the event and p ' is the momentum after the event. Collisions can be very complicated. They can get so complicated that they can’t really be solved as a matter of fact. To simplify things, we will deal with collisions involving two bodies. We will also deal with one-dimensional collisions (like between balls rolling along a track), or else two dimensional collisions that occur at right angles. This helps out a great deal. Imagine trying to deal with a violent collision between two cars. Before the collision you have two bodies in motion that have momentum. After the collision you’ve got thousands of them – all the bits and pieces of the cars flying off in every imaginable direction. Collisions can be classified according to the energy interaction that takes place: Elastic collision kinetic energy is conserved Inelastic collision kinetic energy is not conserved Perfectly inelastic collision objects stick together and have the same velocity. Application of Conservation of Momentum: On the AP Test, you will only be given the equation for momentum, p = mv. From this you will have to derive the formula for a two-body collision. Here’s how to do it. We begin with a two-body collision where each object has some velocity prior to the collision. Both objects have momentum equal to the mass times the velocity. 135 The total momentum before the collision is: p m1v1 m2v2 After the collision the velocities will have changed. The total momentum will be: p ' m1v1 ' m2v2 ' We know that the momentum before must equal the momentum after: p p' so m1v1 m2v2 m1v1 ' m2v2 ' So here is an equation which will work for any two-body collision: m1v1 m2v2 m1v1 ' m2v2 ' Perfectly Inelastic Collisions: These are collisions where the two objects stick together after they collide. The key thing to remember here is that after the collision both objects stick together and have the same velocity. We see a 15 000 kg railroad car moving along the track at 2.5 m/s. It collides and couples with a stationary 12 500 kg car. What is the new velocity of the two cars? p m1v1 p ' m1v1 ' m2v2 ' The velocity of each is the same (they’ve joined together into a “single body”) so we can simplify the equation for the final momentum: p ' m1 m2 v ' Now we set the two equations equal to each other: m1v1 m1 m2 v ' And we get our equation. Now we solve for the final velocity: m1v1 m 1 m v' 15 000 kg 2.5 1.4 m1 m2 s 15 000 kg 12 500 kg s 2 mud balls collide in a perfectly elastic collision. One has a mass of 4.0 kg, the second has a mass of 3.5 kg. The first one has an initial velocity of 3.4 m/s, the second has an initial velocity of – 4.8 m/s. What is their velocity after the collision? 136 p m1v1 m2v2 p ' m1 m2 v ' m1v1 m2v2 m1v1 m2v2 m1 m2 v ' v' m1 m2 m m 4.0 kg 3.4 3.5 kg 4.8 vf s s 0.43 m 4.0 kg 3.5 kg s The negative sign means that they are going in the opposite direction from what our first mud ball was doing. We chose its direction as the positive direction. Something Completely Different: In 1889, Chris L. Rutt, a newspaperman in St. Joseph, Missouri, began working on creating a self-rising pancake mix. Within a year, he and two associates developed the first pancake mix ever made. But what to call the stuff? Soon after the invention was nailed down, Rutt saw a vaudeville team known as Baker and Farrell. In the act Baker, dressed as a Southern mammy, sang a popular tune called "Aunt Jemima". Inspired by the name and image, Rutt stole them both and called his new product “Aunt Jemima pancake mix”. Rutt sold the company to R.T. Davis Mill and Manufacturing Company, which promoted the new product at the World's Columbian Exposition in Chicago in 1893. The company hired Nancy Green, a famous African- American cook born in Montgomery County, Kentucky, to play the part of Aunt Jemima and demonstrate the pancake mix. As Aunt Jemima, Nancy Green made and served over one million pancakes during the run of the fair. Buyers placed over 50,000 orders for Aunt Jemima pancake mix. Over the next thirty years, Green played the part of Aunt Jemima at expositions all over the country. An involved painting of Nancy Green as a stereotype black mammy was pictured on packages of Aunt Jemima Pancake mix. In 1917, she was redrawn as a smiling, heavyset black housekeeper with a bandanna wrapped around her head. In 1989, the company modernized Aunt Jemima, making her thinner, eliminating her bandanna, and giving her a perm and a pair of pearl earrings. In 1923, Nancy Green died in an automobile accident at the age of 89. 137 Bouncing Collisions: When two objects collide and bounce off each other, there are two possible energy interactions. Kinetic energy can be conserved or not conserved. If it is conserved, then we have a perfectly elastic collision. The other possibility is that some of the kinetic energy is transformed into other forms of energy. This is mostly what happens in the real world. The type of problem you will be tasked with solving will have two objects bouncing off each other. You will know three of the four velocities and solve for the fourth. You may be asked whether kinetic energy is conserved and, if it isn’t, how much kinetic energy is lost. Two balls hit head on as shown, what is the final velocity of the second ball if the first one’s final velocity is –1.50 m/s? v1 = 2.30 m's v2 = 1.30 m's m1 = 1.50 kg m2 = 1.85 kg The general conservation of momentum equation is: m1v1 m2v2 m1v1 ' m2v2 ' m1v1 m2v2 m1v1 ' All we have to do is solve for v2’. v2 ' m2 1.50 kg 2.30 m 1.85 kg 1.30 m 1.50 kg 1.50 m v2 ' s s s 1.78 m 1.85 kg s Two balls roll towards each other and collide as shown. The second’s ball velocity after the collision is 3.15 m/s. (a) What is the velocity of the first ball after the collision? (b) How much kinetic energy is lost during the collision? v1 =4.75 m's v2 = 8.60 m's m1 = 2.50 kg m2 = 2.85 kg m1v1 m2v2 m2v2 ' (a) m1v1 m2v2 m1v1 ' m2v2 ' v1 ' m1 138 2.50 kg 4.75 m 2.85 kg 8.60 m 2.85kg 3.15 m v1 ' s s s 8.64 m 2.50 kg s (b) Difference in kinetic energy before and after collision: 1 1 K m1 v1 m2 v2 2 2 Kinetic energy before the collision: 2 2 2 2 1 m 1 m K 2.50 kg 4.75 2.85 kg 8.60 133.6 J 2 s 2 s 1 1 K ' m1 v1 ' m2 v2 ' 2 2 Kinetic energy after the collision 2 2 2 2 1 m 1 m K ' 2.50 kg 8.64 2.85 kg 3.15 107.5 J 2 s 2 s K K ' K 133.6 J 107.5 J 26.1 J What happened to this energy? Where did it go? Well, it was converted into some other form of energy, most likely heat. Explosions: The final type of collision is the explosion. Not like you get with a bomb. That would be one very hard problem. No, we want us a nice, simple two body explosion. An example you ask? Okay, you stand on a skateboard and throw a bowling ball away from you. This would be a simple two-body explosion. The idea of an explosion is that you have two bodies that are at rest – this means that they have no momentum, don’t it? Anyway, the explosion takes place and the bodies end up moving away from each other. Because momentum is conserved, their final momentum, when added together, must still equals zero; the momentum they started out with. A 23 kg girl is standing on a low friction 2.0 kg cart. She is at rest. She then throws a heavy 8.9 kg rock, giving it a speed of 7.5 m/s. What is the final speed of the girl/cart system? m1v1 m2v2 m1v1 ' m2v2 ' becomes: 0 m1v1 ' m2v2 ' m2v2 ' Solve for the speed of the girl: v1 ' m1 139 m2v2 ' m 1 m v1 ' 8.9 kg 7.5 2.7 m1 s 23 kg 2.0 Kg s The negative sign means that the girl is going in the opposite direction from the rock. AP Test Question: Two identical objects A and B of mass M move on a one-dimensional, horizontal air track. Object B initially moves to the right with speed vo. Object A initially moves to the right with speed 3vo, so that it collides with object B. Friction is negligible. Express your answers to the following in terms of M and vo. a. Determine the total momentum of the system of the two objects. p m1v1 m2v2 M 3v0 Mv0 4 Mv0 b. A student predicts that the collision will be totally inelastic (the objects stick together on collision). Assuming this is true, determine the following for the two objects immediately after the collision. i. The speed ii. The direction of motion (left or right) m1v1 m2v2 m1v1 m2v2 m1 m2 v ' v' m1 m2 M v0 M 3v0 4 Mv0 v' 2 v0 M M 2M Moves right (positive direction). When the experiment is performed, the student is surprised to observe that the objects separate after the collision and that object B subsequently moves to the right with a speed 2.5 vo . c. Determine the following for object A immediately after the collision. i. The speed ii. The direction of motion (left or right) m1v1 m2v2 m2v2 ' m1v1 m2v2 m1v1 ' m2v2 ' v1 ' m1 Mv0 M 3 v0 M 2.5 v0 v1 ' 4 v0 2.5 v0 1.5 v0 M Object A move to the right 140 d. Determine the kinetic energy dissipated in the actual experiment. 1 1 1 2 1 K ' m1 v1 ' m2 v2 ' 2 K m1v12 m2v2 2 2 2 2 2 K 1 2 2 1 M 3v0 Mv0 2 2 1 2 M 9 v0 2 v0 2 5 Mv0 2 1 K ' M 1.5v0 2.5v0 2 2 2 1 M 2.25 v0 2 6.25 v0 2 2 4.25M v0 2 K K ' K 5 Mv02 4.25 Mv02 0.75 Mv02 Glancing collisions: This type of collision takes place in two dimensions. Collisions on a pool table are good examples of this type of collision. Momentum still has to be conserved – it’s the law for crying out loud! But the problems can become fairly complex. One technique to keep track of what’s going on is to break things into x and y components: px px ' p y px ' v1f v1i O O Before v2f After We will deal with greatly simplified collisions. Basically with right angles and one of the bodies at rest at the beginning. An 8.00 kg mass moving east at 15.0 m/s strikes a 10.0 kg mass that is at rest. The 8.00 kg mass ends up going south at 4.00 m/s. (a) What is the velocity of the second ball? 141 v2’ v1 Before v1’ After (a) We analyze the momentum in the x and y directions. The x direction: m1v1x m2v2 x m1v1x ' m2v2 x ' becomes m1v1x m1v2 x ' This is because the second body has no initial velocity so it has no initial momentum in either the x or y direction. After the collision, the first body has momentum only in the y direction, so the x direction momentum after the collision involves only the second body. m1v1x m 1 m v2 x ' 8.00 kg 15.0 12 m2 s 10.0 kg s Now let’s look at the y direction: m1v1y m2v2 y m1v1y ' m2v2 y ' 0 m1v1y ' m2v2 y ' The first body’s initial motion is only in the x direction, therefore it has no initial momentum in the y direction. The second body is at rest at the beginning so the total initial y direction momentum is zero. m 8.00 kg 4.00 m1v1y ' s m m2v2 y ' m1v1 y ' v2 y ' 3.20 m2 10.0 kg s Now we can solve for the velocity of the second body using the Pythagorean theorem. 2 2 2 2 m m m v2 v2 f x v2 f y 12.0 3.20 12.4 s s s 142 Now we find the angle : Trigonometry is just the thing to find the old angle: v2y’ v2’ 3.20 m v2 y ' tan s 14.9o v2x’ v2 x ' m 12.0 s A 2.0 kg block is sliding on a smooth table top. It has a totally inelastic collision with a 3.0 kg block that is at rest. Both blocks, now moving, hit the spring and compress it. Once the blocks come to rest, the spring restores itself and launches the blocks. They slide off the right side of the table. The spring constant is 775 N/m. Find: (a) the velocity of the two blocks after the collision. (b) The distance the spring is compressed. (c) The velocity of the blocks after they leave the spring. (d) The distance the blocks travel before they hit the deck after the leave the tabletop. (e) The kinetic energy of the blocks just before they hit the deck. v = 10.0 m/s 1.2 m (a) m1v1 m1v1 ' m2v2 ' m1v1 m 1 m v' 2.0 kg 10.0 4.0 m1 m2 s 2.0 kg 3.0 kg s (b) 1 2 1 2 mv kx xv m 4.0 m 2.0 kg 3.0 kg 0.32 m 2 2 k s kg m 775 s2 m (c) If energy is conserved in the spring, they should have the same speed when they leave as they had going into the spring. So they should be moving at 4.0 m/s to the right. (d) Find the time to fall: 1 2y 2 1.2 m y at 2 t 0.495 s 2 a m 9.8 2 s 143 m x vt 4.0 0.495 s 2.0 m s (e) The kinetic energy of the blocks at the bottom (just before they hit) must equal the potential energy at the top of table plus the kinetic energy the blocks had before they began to fall. 2 1 m 1 m K ' mgy mv 2 5.0 kg 9.8 2 1.2 m 5.0 kg 4.0 99 J 2 s 2 s Examples of Conservation of Momentum: When you fire a rifle, you experience the consequences of the conservation of momentum. The bullet is fired out of the thing at a very high velocity. Because momentum must be conserved, the rifle must end up going in the opposite direction. We call this backwards motion the “recoil”. Rockets, which we discussed using Newton’s third law, can also be explained using the conservation of momentum. The rocket is an explosion type event. It contains a large amount of fuel. The fuel is ignited and blasts out of the rocket nozzle at a high velocity. The rocket must accelerate in the opposite direction so that momentum can be conserved. Dear Cecil: Cecil, you are my hero. My ultimate goal in life is to be the polymath you are. My question concerns a mythical "chicken gun" used for testing jet engines. I have heard tales of store- bought poultry being shot out of a gun at 500 mph into a running jet engine to test the engine's mettle should a pigeon or some other fowl have the misfortune to cross paths with a 747. Does this gun exist, and how does it shoot a roasting hen at that speed without said bird disintegrating? --NoraPatric, via AOL Cecil replies: One problem with researching this question is that everyone thinks he has to tell you the chicken joke. Seems the French borrowed the chicken gun from an American aircraft company to test the windshields of their high-speed trains. After the first test they called the American engineers and said, "Sacrebleu, ze chicken destroy ze windshield and dent ze back wall! What gives?" Having asked a few questions, the engineers replied, "Next time let the chicken thaw first." Talked to two different guys who swore this really happened. Bet they believe in the $250 Mrs. Fields cookie recipe, too. One of the main users of the chicken gun (also known as the chicken cannon or turkey gun) is Pratt & Whitney, the jet engine manufacturer. The "chicken ingestion test," as it's called, is one of a series of stress tests required by the Federal Aviation Administration before a new engine design can be certified. The tests take place in a concrete building large enough to enclose an entire jet engine. With the engine operating at full speed, the cannon uses compressed air to shoot chicken carcasses (or sometimes duck or turkey carcasses) into the turbine at 180 mph (not 500 mph). This is the approximate speed a plane would be traveling if it encountered a bird during takeoff or landing, when most such incidents occur. The chickens are bought not from the corner grocery but from a game farm; the engineers apparently figure that for maximum realism they'd better use birds with feathers.Bird disintegration occurs only after the chick hits the fan. If the turbine disintegrates too, or if the engine can't be operated safely for another twenty minutes after impact, 144 the design fails the test. Other stress tests involve water and ice. The most pyrotechnic test of all requires that dynamite charges be strapped to the compressor blades and detonated while the engine is going full blast. (Needless to say, this is the last test of the day.) If the exploding blades aren't completely contained by the fan case, it's back to the drawing board. Better to have pieces of engine embedded in the concrete walls of the test building than in some poor passenger's skull. --CECIL ADAMS My Garden IF I could put my woods in song And tell what's there enjoyed, All men would to my gardens throng, And leave the cities void. In my plot no tulips blow,-- Snow-loving pines and oaks instead; And rank the savage maples grow From Spring's faint flush to Autumn red. My garden is a forest ledge Which older forests bound; The banks slope down to the blue lake-edge, Then plunge to depths profound. Here once the Deluge ploughed, Laid the terraces, one by one; Ebbing later whence it flowed, They bleach and dry in the sun. The sowers made haste to depart,-- The wind and the birds which sowed it; Not for fame, nor by rules of art, Planted these, and tempests flowed it. Waters that wash my garden-side Play not in Nature's lawful web, They heed not moon or solar tide,-- Five years elapse from flood to ebb. Hither hasted, in old time, Jove, And every god,--none did refuse; And be sure at last came Love, And after Love, the Muse. Hither hasted, in old time, Jove, And every god,--none did refuse; And be sure at last came Love, And after Love, the Muse. 145 Keen ears can catch a syllable, As if one spake to another, In the hemlocks tall, untamable, And what the whispering grasses smother. Æolian harps in the pine Ring with the song of the Fates; Infant Bacchus in the vine,-- Far distant yet his chorus waits. Canst thou copy in verse one chime Of the wood-bell's peal and cry, Write in a book the morning's prime, Or match with words that tender sky? Wonderful verse of the gods, Of one import, of varied tone; They chant the bliss of their abodes To man imprisoned in his own. Ever the words of the gods resound; But the porches of man's ear Seldom in this low life's round Are unsealed, that he may hear. Wandering voices in the air And murmurs in the wold Speak what I cannot declare, Yet cannot all withhold. When the shadow fell on the lake, The whirlwind in ripples wrote Air-bells of fortune that shine and break, And omens above thought. But the meanings cleave to the lake, Cannot be carried in book or urn; Go thy ways now, come later back, On waves and hedges still they burn. These the fates of men forecast, Of better men than live to-day; If who can read them comes at last He will spell in the sculpture,'Stay.' --- Ralph Waldo Emerson 146