Transfer of Momentum The Law of Conservation of Momentum Momentum can be transferred when objects collide. The objects exert equal and opposite forces on each other, causing both objects to change velocity. Before After The ball on the left transfers its momentum thru the three middle balls to the ball on the right. The balls in the middle do not move. When a car is hit from behind it lunges forward because momentum is transferred from the car in the back. Momentum is Conserved In any interaction (objects colliding or pushing off from each other) momentum is conserved: the total amount of momentum doesn t change. It is just redistributed! before 1 kg 1 kg collision 1 kg 1 kg v = 3 m/s p = 3 kgm/s after 1 kg v = 1 m/s Equal and opposite forces v = 1 m/s + p = 1 kgm/s are applied on each other. p = 1 kgm/s + 1 kg v = 3 m/s p = 3 kgm/s p net = 4 kgm/s Momentum is conserved! p net = 4 kgm/s Law of Conservation of Momentum If there are no external forces, the net momentum of a system remains constant. Law of Conservation of Momentum We know that an impulse can change momentum, so it must be included in our equation. But only an external impulse must be included one that changes the momentum of the system. Σp before ± I = Σp after The equal and opposite forces of a collision cancel each other out, so they don t need to be included. collision External Impulse Any external force changes the total momentum and must be included as an impulse. external force F Positive External Impulse Combined Objects Sometimes objects combine or split. When combined, the mass = m 1 + m 2 = m 1+2. Example 2: A 5 g bullet is shot into a resting 2 kg block. How fast are the two going afterwards? Example 1: A 2 kg mass going 1 m/s is pulled by an 8 N force for 4 sec. How fast is the mass going afterwards? Before m = 2 kg +W 8 N 4 sec After m = 2 kg v = 1 m/s v =? There is only one object, so Σp before = mv and there is an external impulse. Σp before ± I = Σp after mv B + Ft = mv A 2(1)+ 8(4) = 2(v A ) 2 + 32 = 2v A 34 = 2v A v A = 17 m/s 5 g; 4 m/s before m = 2 kg v = m/s after v =? There are two objects before (Σp B = p 1 + p 2 ), no external forces (I = ), and one combined object after (Σp A = p 1+2 ). Σp before ± I = Σp after m 1 v 1B + m 2 v 2B + = (m 1 + m 2 )v A.5(4)+ 2() = (2.5)v A.2 + = 1.5v A m after = m 1+2 = 2.5 kg Combined mass.2/2.5 = v A v A =.1 m/s Thrown, Launched, or Pushed Objects Σp before = m ball = 1 kg v b = -2 m/s Thrown objects start at rest, so v = and Σp before =. Since momentum is conserved, Σp after must still =. So the objects must be moving in opposite directions, with equal amounts of momentum. m skater = 4 kg v S =.5 m/s Σp after = = p ball + p skater tice that the more massive object moves slower. The rocket goes up Rockets (and balloons) move by conservation because of momentum, too. Gases are expelled at a very fast velocity, pushing the rocket the opposite direction. the fuel goes down.
1. p 1 + 2 2. m 1 3. v 2A 4. m 1 + 2 A. Velocity of the second object after a collision. B. Velocity of two combined objects. C. Mass of two objects that are stuck together. D. Momentum of two combined objects. 1. p B + I = p A 2. p 1B + p 2B = p 1A + p 2A 3. p 1B + p 2B = p 1+2A 4. p 1+2B = p 1A + p 2A 5. p 1B + p 2B = A. Two moving objects collide and stop. B. An object is pushed and speeds up. C. Two objects at rest push off. D. Two objects collide and stick. E. A moving object breaks apart. 5. v 1 + 2 E. Mass of the first object. 6. = p 1A + p 2A F. Two objects collide and don t connect. A. An object going 3 m/s is pushed by a force for 2 seconds.. B. A cannon shoots a cannonball. C. Two pool balls collide and bounce off of each other. D. An object at rest is pushed by a force. E. A moving object is stopped by a force. F. A person jumps into a boat that is at rest to begin with. G. Two ice skaters push off from each other. H. A moving object explodes into two pieces. Conservation of p Equation: Conservation of p Equation: A 6 kg object going 3 m/s hits a 4 kg object at rest. If the 6 kg object is going 1 m/s afterwards, what is the 4 kg object s final velocity? When is momentum not conserved? A person shoots a bullet from a gun. A) What happens to the gun and shoulder holding the gun? B) How much does the shooter move? A 1 kg object going 3 m/s is pushed by a 12 N force for 4 seconds. Find its final velocity. C) If the bullet hits a person-size target, how much will the target move? D) In the movies a bullet causes a person to fly backwards violently. Explain why this is impossible. A 1, kg cannon shoots a 2 kg cannonball 5 m/s to the right. How fast does the cannon move? How does a rocket move? As a person jumps up, what happens to the earth? A 6 kg person running 1.5 m/s jumps into a 12 kg boat that is at rest. How fast is the boat and person moving afterwards? A 6 kg object moving 1 m/s to the right splits into two equal pieces. If afterwards, one of the pieces is moving 4 m/s to the right, how fast is the other piece moving?
Types of Collisions/ Impulse Graphs Elastic Before There are different types of collisions. Sometimes energy is lost. Sometimes the objects stick together. Yet, in all collisions, momentum is conserved. Just as the word elastic implies, in elastic collisions objects collide, stay separate (bounce off), and all of the kinetic energy is conserved (ΣEk before = ΣEk after ). After Newton s cradle is an obvious example of an elastic collision. You can tell that kinetic energy is conserved because both the left and right ball rise to the same height, v = 3 m/s p = 6 kgm/s Ek = ½(2)3 2 = 9 J before 2 kg 2 kg v = 1 m/s p = 2 kgm/s Σp B = 8 kgm/s ΣEk B = 1 J Ek = ½(2)1 2 = 1 J Conserved Conserved v = 1 m/s p = 2 kgm/s Ek = ½(2)1 2 = 1 J after 2 kg 2 kg v = 3 m/s p = 6 kgm/s Σp A = 8 kgm/s ΣEk A = 1 J Ek = ½(2)3 2 = 9 J Inelastic Where does the energy go? From the Law of Conservation of Energy, you know that energy cannot be lost. Instead it is turned into sound (a crash), damaged objects, or heat. In inelastic collisions the two objects collide, stay separate, but kinetic energy is not conserved. The damage to the cars proves that E k is not conserved. Perfectly Inelastic In perfectly inelastic collisions the two objects collide and stick together. A football player catching a ball, a person jumping into a boat, a bullet shot into a target: all are perfectly inelastic, since two object become one. Type of collision Momentum Kinetic Energy Objects Combine? Elastic Conserved Conserved (ΣEk B = ΣEk A ) Inelastic Conserved t conserved Perfectly Inelastic Conserved t conserved Impulse Graphs Impulse graphs show the force applied to an object over time. Do not mistake the shape of an impulse to mean the direction the object is moving, because it shows force, not distance. 24 21 18 15 12 9 6 3 A Force vs. Time B 1 2 3 4 5 6 7 Area A (triangle) = ½(base)(height) = ½(2)(15) = 15 kgm/s Area B (rectangle) = (base)(height) = (2)(15) = 3 kgm/s Area C (triangle) = ½(3)(15) = 22.5 kgm/s Total Area = 15 + 3 + 22.5 = 67.5 kgm/s Impulse Graph 1 = 67.5 kgm/s Graph 1 C Graph 1 shows a positive impulse: all forces are positive. The acceleration is positive everywhere, too, but decreasing along line C. Graph 2 shows a negative impulse. All forces are negative, so the acceleration is negative everywhere. Along line F, the force is still negative, but just less so. Area = Impulse I = Ft, but on these graphs F is not constant, so you must find the area of the graph. If the force is below the graph, then the area is negative. Negative area = negative impulse. Graph 2 Force vs. Time 1 2 3 4 5 6 7-2 -4-6 -8-1 D E Area D (triangle) = ½(2)(-6) = -6 kgm/s Area E (rectangle) = (3)(-6) = -18 kgm/s Area F (triangle) = ½(2)(-6) = -6 kgm/s Total Area = -6-18 -6 = -3 kgm/s Impulse Graph 2 = -3 kgm/s p = -3 kgm/s The object experiences a negative v. F
Elastic (E), Inelastic (I), or Perfectly Inelastic (P)? (There can be more than one.) 1. Two objects combine together. 7. With Velcro. 2. Momentum is conserved. 8. Σp before = Σp after, and m after = m 1+2 3. Σp before = Σp after, ΣE kbefore ΣE kafter 9. If there is very loud sound, but they don t combine. 4. Kinetic energy is conserved. 1. Kinetic energy is lost. 5. Pool balls hitting each other. 11. Newton s cradle 6. If the objects don t combine, but are damaged. 12. An arrow shot at a target. before after 2 kg 4 kg 2 kg 4 kg 5 m/s 3 m/s v =? 1.67 m/s 13. Calculate the final velocity of the 2 kg object. 14. Calculate ΣEk before. 19. Inelastic collisions do not conserve kinetic energy. Does this violate the Law of Conservation of Energy? 2. Why or why not? 21. If an object explodes when it collides, kinetic energy will increase. Is this elastic or inelastic? 22. Where does the extra energy come from? 15. Calculate ΣEk after. 23. A 1 kg mass going 4 m/s to the right hits a 6 kg object going 2 m/s left. The 1 kg ends up going 2.5 m/s left. A) Find the velocity of the 6 kg object afterwards. 16. Was kinetic energy conserved in the collision? 17. What kind of collision was it? 18. Do expect that the objects were damaged? B) Was the collision elastic or inelastic? Force vs. Tim e Graph 1 2 4 6 8 1 12 14-2 -4 A C -6-8 B -1 Graph 1 or 2? 24. Shows + acceleration? 25. Shows + Impulse? 26. Has a negative area? 27. The object will slow down in the positive direction. 28. Will slow down an object moving to the left. Which force? 29. Has the greatest + force? 3. Shows a weakening negative force? 31. Shows a constant + force? 32. Shows an object speeding up in the + direction. 33. Shows a decreasing + acceleration. 12 1 8 6 4 2 D Force vs. Time 1 2 3 4 5 6 7 8 9 1 11 12 E Graph 2 34. Find the impulse of Graph 1. 35. Find the impulse of Graph 2. 36. If a 2 kg object going 4 m/s feels the impulse in Graph 2, calculate its final velocity.
Momentum In Class Review A. p 1+2B = p 1A + p 2A B. p B I = C. = p 1A + p 2A D. p B + I = p A E. p 1B + p 2B = p 1A + p 2A F. p 1B + p 2B = p 1+2A 1. A car speeds up. 2. A person running catches a football. 3. Two moving cars hit and bounce off. 4. A moving airplane drops a bomb. 5. A rocket at rest turns on its engine: hot gases go back; the rocket goes forward. 6. A moving car uses its brakes to stop. 7. Which has more momentum? A. A fast baseball or a slow baseball? B. A bowling ball or a baseball with the same speed? C. A slow ping pong ball or a house? 8. Give two ways momentum can change. 9. Does a large force always cause a large impulse? Explain. 1. 15 N acts for 8 seconds. How much momentum was gained? m = 4 kg p before = 12 kgm/s 8 N m = 4 kg p after = 36 kgm/s 16. Elastic, Inelastic, or Perfectly Inelastic (could be more than one)? A. Σp before = Σp after, ΣE kbefore ΣE kafter B. Σp before = Σp after, ΣE kbefore = ΣE kafter 11. How much momentum was gained above? 12. How big is the impulse acting on the object? 13. Calculate the time the force acted. 14. Calculate the acceleration of the object. 15. What is the final velocity of the object? C. Σp before = Σp after, and m after = m 1+2 D. There is little or no sound. E. There is a lot of noise. F. The objects are mangled, or crushed. 6 kg 4 kg 17. Two objects collide as shown above. A. What happens to the momentum of the 4 kg object? B. What happens to the momentum of the 6 kg object? C. What happens to the total momentum of the system? 6 m/s Cart 1 m/s.5 sec 6 m/s Cart 2.1 sec 18. Two identical carts moving 6 m/s stop. The Cart 1 hits a spring. The Cart 2 just hits a wall. A. Calculate the initial momentum of the carts. B. Calculate the change of momentum of the carts. C. Which cart experienced the bigger change of momentum? D. Which cart felt the bigger impulse? E. Which cart felt the bigger force? F. Calculate the force on each cart. G. So, to give the same p you have two choices:
2,5 kg 8 m/s A. What is the mass of the ship? B. What is the weight of the ship? C. Calculate the final velocity of the ship. 19. Slim Jim is also an astronaut. His space ship Galactic Cruiser is at rest when he shoots his space cannon. D. Which has more momentum afterwards: the ship or the projectile? 2. A 2 g bullet is shot 8 m/s into 5 kg object. What is the final speed of the combined object? A. If 1 g = 1 kg, what is the mass of the bullet in kilograms? 2 g 8 m/s m/s 5 kg v =? 5 kg B. What is the mass of the combined object? C. Under the diagram, calculate the final speed. D. The numbers given are realistic for a bullet and a person. In movies, a bullet causes a person to be thrown backwards violently. How likely is the movie scenario? Explain. Graph 1 1 9 8 7 6 5 4 3 2 1 A Force vs. Time B.5 1 1.5 2 2.5 3 3.5 4 4.5 5 C Graph 2 Force vs. Time 2.5 5 7.5 1 12.5 15 17.5 2-1 -2-3 -4-5 -6-7 -8-9 -1 D E 21. Use the graphs above to answer the following questions. 22. Graph 1 or Graph 2? A. Shows an object with a positive acceleration B. Could be an object moving to the right and slowing down. C. Shows a negative change of speed. D. Shows a force pushing to the left. 23. Force A, B, C, D, or E (could be more than one)? A. Is the strongest positive force. B. Is the greatest negative force. C. Is the weakest positive force. D. Will cause the fastest negative acceleration. E. Is the strongest force pulling left. F. Shows negative acceleration. 24. Find the impulse of Graph 1. 25. If a 2 kg object going 6 m/s feels the impulse on Graph 1, find its final velocity.
Momentum Flow Chart Momentum distills down to three main concepts: 1) An impulse causes a change in an object s momentum: I = F t = p; 2) Momentum is conserved when objects interact: Σpbefore = Σpafter; 3) In elastic collisions, kinetic energy is conserved: ΣEkbefore = ΣEkafter. How many objects? Multiple START momentum before: = p1a + p2a Examples: explosions; thrown or launched objects; objects pushing off from each other. Use combined mass beforehand: p(1+2)b = p1a + p2a Are they moving beforehand? Do the objects interact (collide or push off) with each other? Are they combined beforehand? Collisions Are they combined afterward? Use: p1b + p2b = p1a + p2a Is Ek conserved? (Does ΣEkb = Eka?) Perfectly Elastic Collision: p1b + p2b = p1a + p2a And: EK1b + EK2b = EK1a + EK2a Must do both calculations to decide if it is elastic! Elastic Collision: p1b + p2b = p1a + p2a But: EK1b + EK2b EK1a + EK2a Examples: objects that collide and don t stick together. Single Is there a force or a change of velocity? Use impulse: p = F t Is the object moving? Calculate net momentum: Σp = p1 + p2... momentum. Calculate momentum: p = mv Perfectly Inelastic Collision (Ek not conserved): Use: p1b + p2b = p(1+2)a Examples: objects that collide and stick together. NOTE: For events in multiple dimensions (like explosions), the above flow chart must be followed independently for each dimension. x direction: Σpxbefore = Σpxafter y direction: Σpybefore = Σpyafter. (etc.)
Types of Collisions/ Impulse Graphs Type 1 - Two objects push apart from each other. (Variations: Something is thrown, launched, or shot; a person starts moving on a boat (or rolling object); a rocket or a balloon.) te: In all of these examples there are two objects at rest before that push off of each other (Newton s 3 rd law). After the push they are both moving in opposite directions. Equation: = p 1a + p 2a Ex1 -. Two people on ice skates push against each other. The person on the left is 5 kg, the person on the right is 6 kg. If the person on the right ends up going 3 m/s, how fast is the other person going? Type 2 - Two objects collide and do not stick together. (Variations: Only one of the objects is moving before or after; head on collisions; rear end collisions; a projectile shot through an object.) te: The direction of the object doesn t matter before or after or if one of them is at rest, just be sure to put in a negative velocity for those moving left and a zero velocity for those at rest. Equation: p 1b + p 2b = p 1a + p 2a Ex2 - A 6 kg object going 3 m/s to the right hits a 4 kg object going 4 m/s to the left. If afterward the 6 kg object ends up going 4 m/s to the left, find out what happens to the 4 kg object. Type 3 - Two objects collide and stick together. (Variations: Only one of the objects is moving before or after; head on collisions; rear end collisions; a projectile shot into an object.) te: Perfectly inelastic collisions. Only difference between this and Case 2 is that in this case there is only one moving object afterwards, so combine the masses. The velocity will be the same for both.) Equation: p 1b + p 2b = p a1+2 Ex3 - A.25 kg arrow going 35 m/s is shot into a 6 kg target. The target is on wheels, so find how fast it will roll backwards. Type 4 - An object splits apart. (Variations: explosions; two objects together and moving push off from each other.) te: Simply the opposite of Case 3, if the split occurs only in one direction. If it happens two dimensionally (as in an explosion), you have to do Conservation of Momentum in each direction independently. In the second dimension, the original momentum was zero, so it is actually just Case 1. Equations: x-dir: p b1+2 = p 1a + p 2a ; y-dir: = p 1a + p 2a Ex4 - A 2 kg car is rolling at 1 m/s when its fuel tank explodes into 4 pieces. If 4 kg of it goes forward at 25 m/ s, 8 kg goes backwards, 3 kg goes to the left at 1 m/s, and a piece goes to the right. Find how much mass is going to the right and how fast it is moving. Also find out how fast the 8 kg piece is moving backwards. (Solutions on back)
Type 1 - Two objects push apart from each other. Ex 1 - Two people on ice skates push against each other. The person on the left is 5 kg, the person on the right is 6 kg. If the person on the right ends up going 3 m/s, how fast is the other person going? Equation: = p 1a + p 2a Type 2 - Two objects collide and do not stick together. Ex2 - A 6 kg object going 3 m/s to the right hits a 4 kg object going 4 m/s to the left. If afterward the 6 kg object ends up going 4 m/s to the left, find out what happens to the 4 kg object. Equation: p 1b + p 2b = p 1a + p 2a Type 3 - Two objects collide and stick together. Ex3 - A.25 kg arrow going 35 m/s is shot into a 6 kg target. The target is on wheels, so find how fast it will roll backwards. Equation: p 1b + p 2b = p a1+2 Type 4 - An object splits apart. Ex4 - A 2 kg car is rolling at 1 m/s when its fuel tank explodes into 4 pieces. If 4 kg of it goes forward at 25 m/ s, 8 kg goes backwards, 3 kg goes to the left at 1 m/s, and a piece goes to the right. Find how much mass is going to the right and how fast it is moving. Also find out how fast the 8 kg piece is moving backwards. Equations: x-dir: p b1+2 = p 1a + p 2a ; y-dir: = p 1a + p 2a