1 kg. 10,000 kg. 1 Page. Momentum is a vector so it has a magnitude and a velocity. Its magnitude is the product of its mass and velocity, p = mv.

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1 Momentum The momentum of a single object is simply equal to the product of its mass and its velocity. The symbol for momentum is p. Since mass is a scalar and velocity is a vector, momentum is also a vector. The direction of momentum is always the same as that of the object s velocity. p = mv Momentum is a vector so it has a magnitude and a velocity. Its magnitude is the product of its mass and velocity, p = mv. Its direction is the same as the direction of its velocity. Units of Momentum The unit of momentum can be derived from the above equation. p = mv The SI units of mass are kilograms (kg) and of velocity are meters / second (m/s). Therefore, the units of momentum are kg m/s. There is no special name for the unit of momentum. TRUCK STOP Conceptually, think of momentum as inertia in motion. Newton s first law explains that objects in motion want to stay in motion. But just how much do moving objects wish to stay in motion? Does a 1 kg skateboard moving at 10 m/s want to stay in motion as much as a 10,000 kg truck moving at the same velocity? To answer, think about which one would be harder to stop. Better yet, which one would you rather have approaching you? 10,000 kg 1 kg 1 Page

2 Practice Questions and Problems Example 1: A 20 kg object has a velocity of +4.5 m/s. What is its momentum? p = mv p = (20kg) (4.5 m/s) p = 90 kg m/s p = +90 kg m/s Example 2: A 60 kg object has a velocity of -1.5 m/s. What is its momentum? Momentum of a Single Object Classwork 1. What is the momentum of a 3000 kg truck traveling at -25 m/s? 2. A 1500 kg ferryboat has a momentum of kg m/s. What is the velocity of the ferryboat? 3. A car travels at a constant velocity of +24 m/s and has a momentum of kg m/s. What is the mass of the car? 4. An 8 kg bowling ball rolls at a constant velocity of +3 m/s. What is the momentum of the ball? 5. A bicyclist travels at a constant velocity of +7 m/s. What is the total mass of the bicycle and the bicyclist, when the total momentum is +490 kg m/s? Homework 6. A 45 kg woman runs at a velocity of -5.6 m/s. What is her momentum? 7. A certain bowling ball rolls at a constant velocity of +5 m/s. If the momentum of the ball is kg m/s, what is its mass? 8. What is the velocity of a 0.25 kg arrow with a momentum of -8 kg m/s? 2 Page

3 Momentum of a Closed System of Objects To determine the momentum of a system that has more than one object you have to add together the momenta of all the individual objects. psystem = p1 + p2 + p3 + psystem = m1v1 + m2v2 + m3v3 + Or psystem = Σp psystem = Σmv So if a system were comprised of the objects in Example 1 and 2, above, then the momentum of that total system would simply be the sum of those two momenta. Example 3: Determine the momentum of a system that consists of the two objects from Example 1 and 2, above. psystem = Σp psystem = p1 + p2 psystem = (90 kg m/s) + (-90 kg m/s) psystem = 0 Example 4: Determine the momentum of a system that consists of the two objects. One object, m1, has a mass of 2 kg and a velocity of +5 m/s and the second object, m2, has a mass of 20 kg and a velocity of +3 m/s. psystem = Σp psystem = p1 + p2 psystem = m1v1 + m2v2 psystem = (2 kg) (+5 m/s) + (20 kg)(+3 m/s) psystem = (10 kg m/s) + (60 kg m/s) psystem = 70 kg m/s psystem = +70 kg m/s Example 5: Determine the momentum of a system that consists of the two objects. One object, m1, has a mass of 4 kg and a velocity of +17 m/s in the positive direction and the second object, m2, has a mass of 70 kg and a velocity of +4 m/s. 3 Page

4 Momentum of a System of Objects Class Work 1. Determine the momentum of a system that consists of two objects. One object, m 1, has a mass of 6 kg and a velocity of +13 m/s and a second object, m 2, has a mass of 14 kg and a velocity of +7 m/s. 2. Determine the momentum of a system that consists of two objects. One object, m 1, has a mass of 6 kg and a velocity of +13 m/s and a second object, m 2, has a mass of 14 kg and a velocity -7 m/s. 3. Determine the momentum of a system that consists of three objects. One object, m 1, has a mass of 7 kg and a velocity +23 m/s, a second object, m 2, has a mass of 9 kg and a velocity +7 m/s and the third object, m 3 has a mass of 5 kg and a velocity of -42 m/s. Homework 4. Determine the momentum of a system of the two objects. One object, m 1, has a mass of 35 kg and a velocity of +3.7 m/s and the second object, m 2, has a mass of 57 kg and a velocity of +4.3 m/s. 5. Determine the momentum of a system of the two objects. One object, m 1, has a mass of 35 kg and a velocity of +3.7 m/s and the second object, m 2, has a mass of 57 kg and a velocity of -4.3 m/s. 6. Create your own example of determining the momentum of a system of two objects. Give it to a classmate to solve then provide them with feedback. 4 Page

5 Conservation of Momentum: p before=p after If two objects start at rest then the total momentum of the system before and after is 0 thus: m 1v 1=m 2v 2 Example 6: Determine the momentum of a system that consists of two objects. If the velocity of cart A is 0.33 meter per second after the spring is released, what is the approximate velocity of cart B after the spring is released? Example 7: A 1.2-kilogram block and a 1.8-kilogram block are initially at rest on a frictionless, horizontal surface. When a compressed spring between the blocks is released, the 1.8-kilogram block moves to the right at 2.0 meters per second, as shown. What is the velocity of the 1.2-kilogram block after the spring is released? 5 Page

6 Example 8: The diagram below shows a 4.0-kilogram cart moving in the positive direction and a 6.0-kilogram cart moving in the negative direction on a horizontal frictionless surface. When the two carts collide they lock together. What is the magnitude of the total momentum of the two-cart system after the collision? How fast do you think the two cars move after the collision? 6 Page

7 Collisions Elastic When an elastic collision occurs, objects bounce off each other with no loss in the total kinetic energy of the system. The collision between billiard balls is very close to a perfectly elastic collision. m1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f Inelastic In an inelastic collision objects change shape or stick together. (An egg dropped on the floor) m 1 v 1 + m 2 v 2 = (m 1 + m 2 )v f In an elastic collision the objects have different velocities before and after the collision. Perfectly inelastic collisions will have different starting velocities but the same final velocity, as they are stuck together and moving as one object. All momentum problems use the same equation: Σp i = Σp f which translates into: 1. Elastic Collisions: m 1 v 1 + m 2 v 2 = m 1 v 1f + m 2 v 2f 2. Inelastic Collisions: m 1 v 1 + m 2 v 2 = (m 1 + m 2 ) v f m 1 = mass of 1 st object m 2 = mass of 2 nd object v 1 = velocity of 1 st object v 2 = velocity of 2 nd object v f = final velocity of both objects v 1f = final velocity of 1 st object v 2f = final velocity of 2 nd object 7 Page

8 To solve momentum problems, always organize your information first. The best thing to do is to draw a sketch, and label the masses and velocities. Then, determine whether the collision is inelastic or elastic. Example: A 13,500 kg railroad freight car travels on a level track at a velocity of 4.5 m/s. It collides and couples with a 25,000 kg second car, initially at rest and with brakes released. What is the velocity of the two cars after collision? Step 1: Draw a sketch and label the masses and velocities. Step 2: Determine whether the collision is inelastic or elastic. The problem states the two freight cars couple or stick together which would make it an inelastic collision thus: m 1 v 1 + m 2 v 2 = (m 1 + m 2 ) v f Conservation of Momentum and Perfectly Inelastic Collisions Classwork 7. A 10,500 kg railroad freight car travels on a level track at a velocity of 2.5 m/s. It collides and couples with a 30,500 kg second car, initially at rest and with brakes released. What is the velocity of the two cars after collision? 8. A 40 kg girl skates at 3.5 m/s on ice toward her 65 kg friend who is standing still, with open arms. As they collide and hold each other, what is the velocity of the couple? Elastic Collision Problems Classwork 9. A ball of mass 0.34 kg moving with a velocity of 2.7 m/s to the right collides head on with a 0.24 kg ball at rest. If the collision is elastic, and after the collision the 0.34 kg ball moves with a velocity of 0.46 m/s what is the velocity and direction of the second ball? 10. An ice puck of mass 0.54 kg moving with a velocity of 5m/s to the right collides with a 0.28 kg piece of ice moving with a velocity of 4.2 m/s to the right. If the collision is elastic, and the piece of ice moves with a velocity of m/s after the collision what is the velocity and direction of the ice puck after the collision? 11. A ball of mass 6.5 kg moving with a velocity of 15 m/s to the right collides head-on with a 3.5 kg ball which is at rest. If the collision is elastic, and the 6.5 kg ball moves with a velocity of 4.5 m/s after the collision what is the velocity and direction of the 3.5 kg ball after the collision? 8 Page

9 Notes: Impulse What does Newton s Second Law have to do with momentum? An object with can be stopped if a is applied against it for a given amount of time. A acting for a given amount of time will change an object's momentum. Put another way, an unbalanced force always accelerates an object - either velocitying it up or slowing it down. If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force velocitys the object up. Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed. In a collision, an object experiences a force for a specific amount of time which results in a change in momentum. The result of the force acting for the given amount of time is that the object's mass either velocitys up or slows down (or changes direction). The impulse experienced by the object equals the change in momentum of the object. What is the impulse-momentum change equation? 9 Page

10 Change of Momentum and Impulse Classwork 12. *A 1200 kg car accelerates from 13 m/s to 17 m/s. Find the change in momentum of the car. 13. *A small object with a momentum of 6 kg m/s to the west approaches head-on a large object at rest. The small object bounces straight back with a momentum of 5 kg m/s. What is the change in the momentum of the small object? What is the impulse exerted on the small ball? What is the impulse exerted on the large object? 14. *A 0.03 kg golf ball is hit off the tee at a velocity of 34 m/s. The golf club was in contact with the ball for s. What is the average force on ball by the golf club? 15. *A toy rocket achieves a velocity of 55 m/s after 3 s, when fired straight up. If the average force exerted by the engine is 28 N, what is the toy s mass? 16. *A kg air jet accelerates from rest to 45 m/s before it takes off. What is the change in momentum of the jet? Homework 17. *A 0.17 kg hockey puck slows down from 54 m/s to 35 m/s when it slides on horizontal ice surface. Find the change in momentum of the puck? 18. *A 0.01 kg bullet is fired at 250 m/s into a wooden block that is fixed. The bullet emerges from the block with a velocity of 120 m/s. What is the change in momentum of the bullet? 19. *A 0.05 kg tennis ball moves at a velocity of 10 m/s and is struck by a racket causing it to rebound in the opposite direction at a velocity 16 m/s. What is the change in momentum of the ball? What is the impulse exerted on the ball? What is the impulse exerted on the racket? The diagram to the right depicts the before- and after-collision velocitys of a car that undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In Case B, the car crumples up and sticks to the wall. a. In which case (A or B) is the change in velocity the greatest? Explain. b. In which case (A or B) is the change in momentum the greatest? Explain. c. In which case (A or B) is the impulse the greatest? Explain. 10 Page