Chapter 7 Impulse and Momentum What is momentum? Mass in Motion. p mv How hard something is to stop. hard to stop small mass, big velocity p = m v hard to stop, big mass, small velocity P = M v Hard to Stop! MV STOP IT! Dp = F Dt = = F Dt What is Impulse? Impulse changes momentum. Dv F ma m Dt FD t m D v Dmv I F D t D p Stop it! Hay stack or Concrete Wall? Dp = F Dt = = F Dt Same Impulse & Change in Momentum Different Force, Different Time Dp = F Dt = = F Dt Stop It! Same Impulse, different Force and Time. 7.1 The Impulse-Momentum Theorem There are many situations when the force on an object is not constant. Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. 7.1 The Impulse-Momentum Theorem DEFINITION OF IMPULSE The impulse of a force is the product of the average force and the time interval during which the force acts: J FDt Impulse is a vector quantity and has the same direction as the average force. newton seconds (N s) Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. 7.1 The Impulse-Momentum Theorem J FDt Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. 7.1 The Impulse-Momentum Theorem DEFINITION OF LINEAR MOMENTUM The linear momentum of an object is the product of the object’s mass times its velocity: p mv Linear momentum is a vector quantity and has the same direction as the velocity. kilogrammeter/second (kg m/s) Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. 7.1 The Impulse-Momentum Theorem v v a f o Dt F ma mv mv F f o Dt FDt mvf mvo Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. What is the average Force exerted on a 0.140 kg baseball by a bat given that the ball’s initial velocity is 45.0m/s and its final velocity, after 1.3ms impact, is 65.0m/s in the opposite direction? pmv00 (.14 kg )( 45 ms / ) 6.3 kgms / pff mv (.14 kg )(65 ms / ) 9.1 kgms / Dpp f p0 9.1 kgms / ( 6.3 kgms / ) Dp 15.4 kg m / s I Dp (15.4kg m / s ) 4 F 1.18xN 10 Dt 1.3xs 103 + Bug Splat Compared to the FORCE that acts on the bug, how much force acts on the bus? More Same Less Newton’s 3rd Law: FFbus bug bug bus Bug Splat Compared to the IMPULSE on the bug, how much impulse is on the bus? More Same Less FFbus bug bug bus Fbus bugD t F bug bus D t Bug Splat Compared to the IMPULSE on the bug, how much impulse is on the bus? More Same Less FFbus bug bug bus Impulsebug -Impulse bus Bug Splat Compared to the Change in Momentum of the bug, the momentum of the bus changes? More Same Less Impulse Dp Impulsebug -Impulse bus Dp-bug Dpbus Bug Splat Which undergoes the greater acceleration? F Bug Same Bus a m Which suffers the greatest damage? Bug Same Bus Bounce/ No Bounce Bullets You are at a war protest when the cops start shooting pellets at the rowdy anarchists. Which would you rather be hit with, rubber bullets that bounce or don’t bounce? Both have the same mass & initial speed. Bounce: Dp m( vf ( v0 )) m( v0 v 0 ) 2 mv 0 No Bounce: Dp 0 mv00 mv Newton’s Third Law FFaction reaction Newton’s Third Law DDppFF ballaction cannon reaction DDtt Dppball D cannon ()()mb v bf m b v b00 m c v cf m c v c mb v bf m c v cf m b v b00 m c v c pp Conservation of total,, before total after Momentum! Conservation of Momentum If there are no external forces acting on a system of objects then the total momentum before an event is equal to the total momentum after an event. ppbefore after The system 100kg 100kg pbefore mv 0 mv THE EVENT v p 2 m mv after 2 Conservation Depends on The System 100kg 100kg pbefore mv THE EVENT External v Force! pm after 2 Momentum is NOT conserved! Colliding Systems For momentum to be conserved you must define The System so all the forces are internal to it and there are no net external forces. Include everything that gets hit. Conservation of Momentum If there are no external forces acting on a system of objects then the total momentum before an event is equal to the total momentum after an event. ppbefore after System: You define the system so that momentum is conserved. •No external forces acting on the system •Internal forces cancel out and don’t change momentum •Include all the objects that collide or ‘go bump’ •Always apply conservation laws to a system When you have defined the system so that the net external forces acting is zero, THEN & ONLY THEN can you invoke Conservation of momentum! ONLY AFTER you haveppbefore defined the aftersystem so that the net external forces acting on it is zero, THEN & ONLY THEN can you invoke Conservation of momentum and apply: Conservation of Momentum Momentum is conserved for which system? (a) (b) (c) INCLUDE EVERYTHING THAT GOES BUMP! NOTICE HOW THE INTERNAL FORCES CANCEL! The recoil momentum of a gun that kicks is more than less than the same as the momentum of the bullet it fires. Momentum is conserved for the system including: gun alone gun + man gun + man + bullet Frictionless surface Charlie and his gun, m = 80kg, shoots off a bullet, m =.01kg, straight ahead at 300 m/s. What is Charlie’s (+gun) recoil velocity? 1. Define the System so net external forces is zero. 2. THEN momentum is conserved ppinitial final pcharlie,,,, i p bullet i p charlie f p bullet f 0 ppcharlie,, f gun f ppcharlie,, f bullet f mc v cf m b v bf .01kg mb 300ms / Frictionless surface vvcf bf 80kg mc .04ms / Other Exploding Systems ppinitial final Rocket Thrust 0 Mvrocket mv gas Dprocket Mv ppinitial final Mvrocket mv gas Dpprocket D gas Dpgas mV Rocket Pushes Gas Out. Gas Pushes Back on Rocket. Collision Systems: Total Momentum is Conserved Once the System is properly defined Momentum is conserved for all Collisions. Total momentum before = Total momentum after. Elastic Collisions: Bounce Momentum is transferred! Total momentum before = Total momentum after Inelastic Collisions: Stick Total momentum before = Total momentum after Collision Decision Both trucks have the same speed and mass. After they collide they will: a) Bounce off each other b) Stick and move to the left c) Stick and move to the right d) Stick and stop dead in the center e) Vaporize Inelastic Collisions: Stick Was Momentum Conserved in this case? Depends on the System! Elastic Collisions: Bounce What is the System so momentum is conserved? If the mass is the same what is her final velocity? 1 2 ppinitial final mv1i mv 2 i mv 1 f mv 2 f v1i v 2 i v 1 f v 2 f 2m/s v2f v 1 i v 2 i v 1 f v2 f 2 m / s ( 1 m / s ) ( 1 m / s ) 1 2 v2 f 2/ m s Momentum is Exchanged! Elastic Collisions What if you didn’t know the final velocity of car 1? Could you still find the final velocities? 1 2 Use Conservation of Momentum AND Kinetic Energy! Solve 2 equations for two unknowns: the velocities! pp 1 2 f 0 KEf KE0 Elastic Collisions ppf 0 mv1ff mv 2 mv 10 mv 20 1 2 KEf KE0 12 1 2 1 2 1 2 mv1ff mv 2 mv 10 mv 20 2 2 2 2 Plus lots of Algebra!! Can you use this method to 1 2 solve for Inelastic Collisions? Kinetic Energy is NOT Conserved! Inelastic Collisions Granny (m=50kg) whizzes around the rink at 3m/s and snatches up Andy (m=25kg). What is their final velocity? (Ignore friction) Before After Granny: m=50kg, v =3m/s Inelastic Collisions 0 Andy: m=25kg, v0=0 What is the system? ppinitial final Before After Granny: m=50kg, v =3m/s Inelastic Collisions 0 Andy: m=25kg, v0=0 m1 v 1if() m 1 m 2 v ppinitial final mv11i v f ()mm12 50kg 3 m / s 1 (50kg 25 kg ) 2 2/ms Before After Collisions Momentum is conserved for all Collisions. Momentum is transferred during Collisions. Elastic: The total Kinetic Energy of the system is conserved. Think Bounce! Inelastic: The total Kinetic Energy of the system is not conserved. Think Stick! Why is momentum conserved during Inelastic Collisions but kinetic energy is not? Inelastic Collisions Kinetic Energy is ‘lost’ to heat & other kinds of energy. There is only one kind of momentum. Therefore it is conserved. 7.2 The Principle of Conservation of Linear Momentum Example 6 Ice Skaters Starting from rest, two skaters push off against each other on ice where friction is negligible. One is a 54-kg woman and one is a 88-kg man. The woman moves away with a speed of +2.5 m/s. Find the recoil velocity of the man. Copyright © 2015 John Wiley & Sons, Inc. All rights reserved. 7.2 The Principle of Conservation of Linear Momentum Pf Po m1v f 1 m2v f 2 0 m1v f 1 v f 2 m2 54 kg 2.5m s v 1.5m s f 2 88 kg Copyright © 2015 John Wiley & Sons, Inc.