Linear Momentum and Collisions Chapter 9 Linear Momentum and Collisions

9-1 Linear Momentum Momentum is a vector; its direction is the same as the direction of the velocity.

9-1 Linear Momentum Change in momentum: mv 2mv

ConcepTest 9.2b Momentum and KE II A system of particles is known to have a total momentum of zero. Does it necessarily follow that the total kinetic energy of the system is also zero? 1) yes 2) no

ConcepTest 9.2b Momentum and KE II A system of particles is known to have a total momentum of zero. Does it necessarily follow that the total kinetic energy of the system is also zero? 1) yes 2) no Momentum is a vector, so the fact that ptot = 0 does not mean that the particles are at rest! They could be moving such that their momenta cancel out when you add up all of the vectors. In that case, since they are moving, the particles would have non-zero KE.

ConcepTest 9.2c Momentum and KE III Two objects are known to have the same momentum. Do these two objects necessarily have the same kinetic energy? 1) yes 2) no

ConcepTest 9.2c Momentum and KE III Two objects are known to have the same momentum. Do these two objects necessarily have the same kinetic energy? 1) yes 2) no If object #1 has mass m and speed v and object #2 has mass 1/2 m and speed 2v, they will both have the same momentum. However, since KE = 1/2 mv2, we see that object #2 has twice the kinetic energy of object #1, due to the fact that the velocity is squared.

9-2 Momentum and Newton’s Second Law Newton’s second law, as we wrote it before: is only valid for objects that have constant mass. Here is a more general form, also useful when the mass is changing:

9-3 Impulse Impulse is a vector, in the same direction as the average force.

9-3 Impulse We can rewrite as So we see that The impulse is equal to the change in momentum.

9-3 Impulse

ConcepTest 9.3a Momentum and Impulse 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. Both forces act for the same mount of time. How does the change of the boulder’s momentum compare to the change of the pebble’s momentum? 1) greater than 2) less than 3) equal to

ConcepTest 9.3a Momentum and Impulse 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. Both forces act for the same mount of time. How does the change of the boulder’s momentum compare to the change of the pebble’s momentum? 1) greater than 2) less than 3) equal to

ConcepTest 9.3b Velocity and Impulse 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. Both forces act for the same mount of time. How does the change of the boulder’s velocity compare to the change of the pebble’s velocity? 1) greater than 2) less than 3) equal to

ConcepTest 9.3b Velocity 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 velocity compare to the rate of change of the pebble’s velocity? 1) greater than 2) less than 3) equal to

ConcepTest 9.7 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 9.7 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

Example To make a bounce pass, a player throws a 0.60 kg basketball toward the floor. The ball hits the floor with a speed of 5.4 m/s and rebounds with the same speed and angle. What impulse does the floor deliver to the basketball? If the ball is in contact with the floor for 0.1 s, what average force did the floor exert?

9-4 Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change

9-4 Conservation of Linear Momentum Internal Versus External Forces: Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it.

9-4 Conservation of Linear Momentum Furthermore, internal forces cannot change the momentum of a system. However, the momenta of components of the system may change.

9-4 Conservation of Linear Momentum An example of internal forces moving components of a system:

ConcepTest 9.14a Recoil Speed I Amy (150 lbs) and Gwen (50 lbs) are standing on slippery ice and push off each other. If Amy slides at 2 m/s, what speed does Gwen have? (1) 1 m/s (2) 2 m/s (3) 3 m/s (4) 4 m/s (5) 6 m/s 150 lbs 50 lbs

ConcepTest 9.14a Recoil Speed I Amy (150 lbs) and Gwen (50 lbs) are standing on slippery ice and push off each other. If Amy slides at 2 m/s, what speed does Gwen have? (1) 1 m/s (2) 2 m/s (3) 3 m/s (4) 4 m/s (5) 6 m/s The initial momentum is zero, so the momenta of Amy and Gwen must be equal and opposite. Since p = mv, then if Amy has 3 times more mass, we see that Gwen must have 3 times more speed. 150 lbs 50 lbs

9-5 Inelastic Collisions A completely inelastic collision:

9-5 Inelastic Collisions For collisions in two dimensions, conservation of momentum is applied separately along each axis:

ConcepTest 9.12a Inelastic Collisions I 1) 10 m/s 2) 20 m/s 3) 0 m/s 4) 15 m/s 5) 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

ConcepTest 9.12a Inelastic Collisions I 1) 10 m/s 2) 20 m/s 3) 0 m/s 4) 15 m/s 5) 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)

Example A car (m1 = 950 kg) heading due E at 20.0 m/s collides with a minivan (m2 = 1300 kg) headed due N (see figure.) After the collision, the wreckage moves at 40.0˚ above the x-axis. Find the initial speed of the minivan and the final speed of the wreckage.

9-6 Elastic Collisions In elastic collisions, both kinetic energy and momentum are conserved. One-dimensional elastic collision:

ConcepTest 9.10a Elastic Collisions I Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision? 1) situation 1 2) situation 2 3) both the same v 2 1 at rest

ConcepTest 9.10a Elastic Collisions I Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision? 1) situation 1 2) situation 2 3) both the same v 1 Remember that the magnitude of the relative velocity has to be equal before and after the collision! In case 1 the bowling ball will almost remain at rest, and the golf ball will bounce back with speed close to v. v 2 2v In case 2 the bowling ball will keep going with speed close to v, hence the golf ball will rebound with speed close to 2v.

9-6 Elastic Collisions Two-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object:

Example A 722-kg car stopped at an intersection is rear-ended by a 1620-kg truck moving with a speed of 14.5 m/s. If the car was in neutral and the brakes were off, so that the collision is approximately elastic, find the final speed of both vehicles after the collision.

9-7 Center of Mass The center of mass of a system is the point where the system can be balanced in a uniform gravitational field.

9-7 Center of Mass For two objects: The center of mass is closer to the more massive object.

9-7 Center of Mass The center of mass need not be within the object:

9-7 Center of Mass Motion of the center of mass:

9-7 Center of Mass The total mass multiplied by the acceleration of the center of mass is equal to the net external force: The center of mass accelerates just as though it were a point particle of mass M acted on by

ConcepTest 9.18 Baseball Bat 1) at the midpoint 2) closer to the thick end 3) closer to the thin end (near handle) 4) it depends on how heavy the bat is Where is center of mass of a baseball bat located?

ConcepTest 9.18 Baseball Bat 1) at the midpoint 2) closer to the thick end 3) closer to the thin end (near handle) 4) it depends on how heavy the bat is Where is center of mass of a baseball bat located? Since most of the mass of the bat is at the thick end, this is where the center of mass is located. Only if the bat were like a uniform rod would its center of mass be in the middle.

Summary of Chapter 9 Linear momentum: Momentum is a vector Newton’s second law: Impulse: Impulse is a vector The impulse is equal to the change in momentum If the time is short, the force can be quite large

Summary of Chapter 9 Momentum is conserved if the net external force is zero Internal forces within a system always sum to zero In collision, assume external forces can be ignored Inelastic collision: kinetic energy is not conserved Completely inelastic collision: the objects stick together afterward

Summary of Chapter 9 A one-dimensional collision takes place along a line In two dimensions, conservation of momentum is applied separately to each Elastic collision: kinetic energy is conserved Center of mass:

Summary of Chapter 9 Center of mass:

Summary of Chapter 9 Motion of center of mass: