Chapter 6 Linear Momentum and Collisions © 2010 Pearson Education, Inc.

Units of Chapter 6 Linear Momentum Impulse Conservation of Linear Momentum Elastic and Inelastic Collisions Center of Mass Jet Propulsion and Rockets © 2010 Pearson Education, Inc.

6.1 Linear Momentum Definition of linear momentum: The linear momentum of an object is the product of its mass and velocity. Note that momentum is a vector—it has both a magnitude and a direction. SI unit of momentum: kg • m/s. This unit has no special name. © 2010 Pearson Education, Inc.

6.1 Linear Momentum For a system of objects, the total momentum is the vector sum of each. © 2010 Pearson Education, Inc.

6.1 Linear Momentum The change in momentum is the difference between the momentum vectors. © 2010 Pearson Education, Inc.

6.1 Linear Momentum If an object’s momentum changes, a force must have acted on it. The net force is equal to the rate of change of the momentum. © 2010 Pearson Education, Inc.

Question 6.2a Momentum and KE I A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system? a) momentum of the system is positive b) momentum of the system is negative c) momentum of the system is zero d) you cannot say anything about the momentum of the system Answer: c

Question 6.2a Momentum and KE I A system of particles is known to have a total kinetic energy of zero. What can you say about the total momentum of the system? a) momentum of the system is positive b) momentum of the system is negative c) momentum of the system is zero d) you cannot say anything about the momentum of the system Because the total kinetic energy is zero, this means that all of the particles are at rest (v = 0). Therefore, because nothing is moving, the total momentum of the system must also be zero.

Question 6.2c Momentum and KE III Two objects are known to have the same momentum. Do these two objects necessarily have the same kinetic energy? a) yes b) no Answer: b

Question 6.2c Momentum and KE III Two objects are known to have the same momentum. Do these two objects necessarily have the same kinetic energy? a) yes b) no If object #1 has mass m and speed v and object #2 has mass m and speed 2v, they will both have the same momentum. However, because KE = mv2, we see that object #2 has twice the kinetic energy of object #1, due to the fact that the velocity is squared.

6.2 Impulse Impulse is the change in momentum: Typically, the force varies during the collision. © 2010 Pearson Education, Inc.

6.2 Impulse Actual contact times may be very short. © 2010 Pearson Education, Inc.

6.2 Impulse When a moving object stops, its impulse depends only on its change in momentum. This can be accomplished by a large force acting for a short time, or a smaller force acting for a longer time. © 2010 Pearson Education, Inc.

6.2 Impulse We understand this instinctively—we bend our knees when landing a jump; a “soft” catch (moving hands) is less painful than a “hard” one (fixed hands). This is how airbags work—they slow down collisions considerably—and why cars are built with crumple zones. © 2010 Pearson Education, Inc.

Question 6.3a Momentum and Force A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to Answer: c

Question 6.3a Momentum and Force A net force of 200 N acts on a 100-kg boulder, and a force of the same magnitude acts on a 130-g pebble. How does the rate of change of the boulder’s momentum compare to the rate of change of the pebble’s momentum? a) greater than b) less than c) equal to The rate of change of momentum is, in fact, the force. Remember that F = Dp/Dt. Because the force exerted on the boulder and the pebble is the same, then the rate of change of momentum is the same.

Question 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? a) the beanbag b) the rubber ball c) both the same Answer: b

Question 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? a) the beanbag b) the rubber ball c) 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 I=F*t Follow-up: Which one imparts the larger force to the floor?

A 0.17 kg baseball is thrown with a speed of 38.m/s and it is hit straight back to the pitcher with a speed of 62.m/s. What is the magnitude of the IMPULSE exerted upon the ball by the bat? Answer: 17 N∙s

(a) What average force was exerted on the racket? Jennifer hits a stationary 200. gram ball and it leaves the racket at 40. m/s. If time lapse photography shows that the ball was in contact with the racket for 40. ms: (a) What average force was exerted on the racket? (b) What is the ratio of this force to the weight of the ball? (a) 0.20 kN (b) 1.0 × 102

(a) What is the magnitude of its change in momentum? A 0.32 kg ball is moving horizontally 30. m/s just before bouncing off a wall, thereafter moving 25. m/s in the opposite direction. (a) What is the magnitude of its change in momentum? (b) What percentage of the kinetic energy was lost in the collision? (a) 18. N-s (b) 31.% lost

6.3 Conservation of Linear Momentum If there is no net force acting on a system, its total momentum cannot change. This is the law of conservation of momentum. If there are internal forces, the momenta of individual parts of the system can change, but the overall momentum stays the same. © 2010 Pearson Education, Inc.

6.3 Conservation of Linear Momentum In this example, there is no external force, but the individual components of the system do change their momenta: © 2010 Pearson Education, Inc.

6.3 Conservation of Linear Momentum Collisions happen quickly enough that any external forces can be ignored during the collision. Therefore, momentum is conserved during a collision. © 2010 Pearson Education, Inc.

An empty coal-car (mass 20,000. kg) of a train coasts along at 10. m/s An empty coal-car (mass 20,000.kg) of a train coasts along at 10.m/s. An unfortunate 3000. kg elephant falls from a bridge and drops vertically into the car. Determine the speed of the car immediately after the elephant is added to its contents. 8.7 m/s

6.4 Elastic and Inelastic Collisions In an elastic collision, the total kinetic energy is conserved. Total kinetic energy is not conserved in an inelastic collision. © 2010 Pearson Education, Inc.

6.4 Elastic and Inelastic Collisions A completely inelastic collision is one where the objects stick together afterwards. © 2010 Pearson Education, Inc.

6.4 Elastic and Inelastic Collisions The fraction of the total kinetic energy that is left after a completely inelastic collision can be shown to be: © 2010 Pearson Education, Inc.

6.4 Elastic and Inelastic Collisions For an elastic collision, both the kinetic energy and the momentum are conserved: © 2010 Pearson Education, Inc.

6.4 Elastic and Inelastic Collisions Collisions may take place with the two objects approaching each other, or with one overtaking the other. © 2010 Pearson Education, Inc.

Question 6.12a Inelastic Collisions I a) 10 m/s b) 20 m/s c) 0 m/s d) 15 m/s e) 5 m/s A box slides with initial velocity 10 m/s on a frictionless surface and collides inelastically with an identical box. The boxes stick together after the collision. What is the final velocity? vf vi M Answer: e

Question 6.12a Inelastic Collisions I a) 10 m/s b) 20 m/s c) 0 m/s d) 15 m/s e) 5 m/s A box slides with initial velocity 10 m/s on a frictionless surface and collides inelastically with an identical box. The boxes stick together after the collision. What is the final velocity? The initial momentum is: M vi = (10) M vf vi M The final momentum must be the same!! The final momentum is: Mtot vf = (2M) vf = (2M) (5)

As shown in Fig. 6-2, a car (mass = 1500 kg) and a small truck (mass = 2000 kg) collide at right angles at an icy intersection. The car was traveling East at 20 m/s and the truck was traveling North at 20 m/s when the collision took place. What is the speed of the combined wreck, assuming a completely inelastic collision? 14.3 m/s

In space, a 4. 0 kg metal ball moving 30 In space, a 4.0 kg metal ball moving 30. m/s has a head-on collision with a stationary 1.0 kg second ball. After the elastic collision, what are the velocities of the balls? Answer: The first becomes 18. m/s, and the second acquires 48. m/s; both in the direction the first was moving.

A 50-gram ball moving +10 m/s collides head-on with a stationary ball of mass 100.g. The collision is elastic. What is the speed of each ball immediately after the collision? Answer: -3.3 m/s and +6.7 m/s

6.5 Center of Mass Definition of the center of mass: The center of mass is the point at which all of the mass of an object or system may be considered to be concentrated, for the purposes of linear or translational motion only. We can then use Newton’s second law for the motion of the center of mass: © 2010 Pearson Education, Inc.

6.5 Center of Mass The momentum of the center of mass does not change if there are no external forces on the system. The location of the center of mass can be found: This calculation is straightforward for a system of point particles, but for an extended object calculus is necessary. © 2010 Pearson Education, Inc.

6.5 Center of Mass The center of mass of a flat object can be found by suspension. © 2010 Pearson Education, Inc.

6.5 Center of Mass The center of mass may be located outside a solid object. © 2010 Pearson Education, Inc.

Question 6.19 Motion of CM Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system? a) it does not move b) it moves away from A with speed v c) it moves toward A with speed v d) it moves away from A with speed ½v e) it moves toward A with speed ½v Answer: d

Question 6.19 Motion of CM Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system? a) it does not move b) it moves away from A with speed v c) it moves toward A with speed v d) it moves away from A with speed ½v e) it moves toward A with speed ½v Let’s say that A is at the origin (x = 0) and B is at some position x. Then the center of mass is at x/2 because A and B have the same mass. If v = Dx/Dt tells us how fast the position of B is changing, then the position of the center of mass must be changing like D(x/2)/Dt, which is simply v.

Question 6.20 Center of Mass The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ? a) higher b) lower c) at the same place d) there is no definable CM in this case (1) X CM (2) Answer: a

Question 6.20 Center of Mass The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ? a) higher b) lower c) at the same place d) there is no definable CM in this case The CM of each half is closer to the top of the semicircle than the bottom. The CM of the whole system is located at the midpoint of the two semicircle CMs, which is higher than the yellow line. (1) X CM (2) CM

6.6 Jet Propulsion and Rockets If you blow up a balloon and then let it go, it zigzags away from you as the air shoots out. This is an example of jet propulsion. The escaping air exerts a force on the balloon that pushes the balloon in the opposite direction. Jet propulsion is another example of conservation of momentum. © 2010 Pearson Education, Inc.

6.6 Jet Propulsion and Rockets This same phenomenon explains the recoil of a gun: © 2010 Pearson Education, Inc.

6.6 Jet Propulsion and Rockets The thrust of a rocket works the same way. © 2010 Pearson Education, Inc.

6.6 Jet Propulsion and Rockets Jet propulsion can be used to slow a rocket down as well as to speed it up; this involves the use of thrust reversers. This is done by commercial jetliners. © 2010 Pearson Education, Inc.

Summary of Chapter 6 Momentum of a point particle is defined as its mass multiplied by its velocity. The momentum of a system of particles is the vector sum of the momenta of its components. Newton’s second law: © 2010 Pearson Education, Inc.

Summary of Chapter 6 Impulse–momentum theorem: In the absence of external forces, momentum is conserved. Momentum is conserved during a collision. Kinetic energy is also conserved in an elastic collision. © 2010 Pearson Education, Inc.

Summary of Chapter 6 The center of mass of an object is the point where all the mass may be considered to be concentrated. Coordinates of the center of mass: © 2010 Pearson Education, Inc.