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Chapter 6.1 Pre-Quiz Complete on a half-sheet of paper and turn in once completed. 1. What two previously discussed quantities define momentum? 2. Which has more momentum: a) 18-Wheeler Truck b) Child on a tricycle c) Need more information 3. (True / False) Momentum is a vector Chapter 6 Momentum and Collisions 6.1 Linear Momentum Definition of linear momentum: The linear momentum of an object is the product of its mass and velocity. Momentum is a vector— meaning… • it has both a magnitude and a direction! SI unit of momentum: kg • m/s This unit has no special name Momentum Linebacker: 108 kg Velocity: 2.00 m/s p=m·v p = 108 kg x 2.00 m/s p = 216 kg · m/s 6.1 Linear Momentum For a system of objects, the total momentum is the vector sum of each. 6.1 Linear Momentum The change in momentum is the difference between the momentum vectors. 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. Momentum Linebacker: 108 kg Velocity: 2.00 m/s Initial Momentum: 216 kg·m/s Final Momentum: 0 kg·m/s Time: 0.315 s Fnet = Δp / Δt Fnet = (- 216 kg·m/s) / (0.315 s) Fnet = - 686 N (opposes motion) 6.2 Impulse Impulse is the change in momentum: Typically, the force varies during the collision. 6.2 Impulse Real-world contact times may be very short. Demonstration Time? Yes. MUWAHAHA! 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. 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 also how airbags work—they slow down collisions considerably—and why cars are built with “crumple zones”. Audi Q7 vs Smart Fiat Mercedes C300 Cat Cat 500 ForTwo Kinetic Energy and Momentum • Make an equation for kinetic energy that uses momentum. • Hint: It should start out… KE = 1/2… Momentum Exit Quiz Complete on half-sheet of paper and turn in as you leave class. 1. How do we define momentum? 2. During a collision, what does impulse represent? 3. Explain the effect of changing contact time for a collision. Momentum Practice Page 207-209 Linear Momentum: 1-6*, 8-11, 16, 25** Impulse: 26-31*, 32-34, 41, 46 * Try these first without your notes! ** Kudos if you can solve this! KE = ½ Δp · v KE = ½ Δp2 / m Momentum Warm-Up 1. Find the momentum of a 1200 kg car travelling at 15 m/s. 2. If a soccer player kicks a 0.43 kg soccer ball, giving it a velocity of 26 m/s. What is the average force between their foot and the ball? 3. How is falling 10 feet onto a trampoline, different than falling 10 feet to the ground? (explain in terms of F, p and t) Impulse, Force, and Contact time For any object coming to rest, the impulse is not dependent upon how it stops. Trade-off: Contact time or Average force Mercedes vs. Smart car Before Collision: Red Mercedes C300 Blue Smart Fortwo 1695 kg 730 kg 17.9 m/s East 17.9 m/s West Watch direction! After Collision: Red Mercedes C300 Blue Smart Fortwo 1695 kg 730 kg 8.95 m/s East 2.88 m/s East Find Δp (impulse) for each car 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. 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: 6.3 Conservation of Linear Momentum Before/after any event* the momentum of the objects involved will be conserved. Therefore, the sum of momenta before equals the sum of the momenta after. *event = collision, objects pushing apart, anything that changes one/both/all objects’ velocity. m1 = 1695 kg v1o = 17.9 m/s v1 = 0 m/s m2 = 730 kg v2o = -17.9 m/s v2 = ? m/s Cannon m1 = 100 kg v1 = ? m/s Cannonball m2 = 5.00 kg v2 = 40.0 m/s Zorba the Dog m1 = 156 kg v1o = 4.2 m/s v1 = ? m/s Boat m2 = 140 kg v2o = 0 m/s v2 = ? m/s 6.3 Conservation of Linear Momentum Classifying Collisions • In this class we will examine three types of collisions: • Elastic Collisions • Inelastic Collisions • Perfectly Inelastic Collisions 6.4 Elastic and Inelastic Collisions For an elastic collision, both the kinetic energy and the momentum are conserved: This is the same as saying… 6.4 Elastic and Inelastic Collisions Before After p1o + p2o = p1 + p2 K1o + K2o = K1 + K2 Total momentum before = Total momentum after Total kinetic energy before = total kinetic energy after 6.4 Elastic and Inelastic Collisions Collisions may take place with the two objects approaching each other, or with one overtaking the other. Page 196 Ex 6.11 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions 1. Elastic Collision • Kinetic energy is conserved (Kf = Ki) • Objects retain their initial shapes. 6.4 Elastic and Inelastic Collisions An inelastic collision is exemplary of real-world collisions where kinetic energy is lost. How is energy “lost”? We can still get by using the elastic collision equations/assumptions though because for many small-scale collisions the KE lost is negligible. 6.4 Elastic and Inelastic Collisions A perfectly inelastic collision is one where the objects stick together afterwards. = mv 6.4 Elastic and InelasticpCollisions Everyday collisions are imperfect inelastic collisions. Often times one or both of the objects colliding will be dented/damaged. Such physical changes require work (energy) to be done and thus kinetic energy is not conserved. Kinetic energy may also be lost to things such as friction and sound. Types of Collisions |-------------------------------------- Momentum is always conserved --------------------------------------| Perfectly Elastic partially elastic partially inelastic Perfectly Inelastic |--KE conserved --||---------------------------------------Some KE lost---------------------------------------| Equations for Each Type • Inelastic: m1v1o + m2v2o = m1v1 + m2v2 Two objects collide and do not remain stuck together, therefore all we know is that the overall momentum is conserved. • Perfectly Inelastic: m1v1o + m2v2o = (m1 + m2)v Two objects collide and remain stuck together, so there is only one final velocity (helpful in reducing the # of variables). Equations for Each Type Allows solving for final velocities when both v1 and v2 are unknown. (When you only know the starting info!) Elastic Collision Practice A 120 kg rock is flying through space at 20.0 m/s when it undergoes an elastic collision with another 90.0 kg rock that is travelling at 15 m/s in the same direction. What is the final velocity of each rock? Elastic Collision Practice A 140 kg bumper car is travelling at 4.5 m/s to the East when it experiences a head-on elastic collision with a 160 kg bumper car that was travelling at 6.0 m/s. What is the final velocity of each car? 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. 6.6 Jet Propulsion and Rockets This same phenomenon explains the recoil of a gun: 6.6 Jet Propulsion and Rockets The thrust of a rocket works the same way. 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. 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. 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. 6.5 Center of Mass The center of mass of a flat object can be found by suspension. 6.5 Center of Mass The center of mass may be located outside a solid object. 6.5 Center of Mass Top Mass = 130 g Loc = (0.0 m, 0.3 m) Middle Mass = 70.0 g Loc = (0.0 m, 0 m) XCM Bottom Mass = 80.0 g Loc = (0.0 m, -0.30 m)