Physics 207 Lecture 12 Oct 13 Goals Chapter 9 Momentum Impulse Solve problems with 1D and 2D Collisions Solve problems having an impulse Force vs time Chapter 10 Understand the relationship between motion and energy Define Potential Kinetic Energy Develop and exploit conservation of energy principle Assignment HW5 due Wednesday HW6 available today For Wednesday Read all of chapter 10 Physics 207 Lecture 12 Pg 1 Inelastic collision in 1 D Example A block of mass M is initially at rest on a frictionless horizontal surface A bullet of mass m is fired at the block with a muzzle velocity speed v The bullet lodges in the block and the block ends up with a speed V In terms of m M and V What is the momentum of the bullet with speed v x v V before after Physics 207 Lecture 12 Pg 2 Inelastic collision in 1 D Example What is the momentum of the bullet with speed v mv Key question Is x momentum conserved Before aaaa After mv M 0 m M V v V after Physics 207 Lecture 12 Pg 3 x before Exercise Momentum Conservation Two balls of equal mass are thrown horizontally with the same initial velocity They hit identical stationary boxes resting on a frictionless horizontal surface The ball hitting box 1 bounces elastically back while the ball hitting box 2 sticks Which box ends up moving fastest A B C Box 1 Box 2 same 1 2 Physics 207 Lecture 12 Pg 6 Exercise Momentum Conservation The ball hitting box 1 bounces elastically back while the ball hitting box 2 sticks Which box ends up moving fastest Notice the implications from the graphical solution Box 1 s momentum must be bigger because the length of the summed momentum must be the same The longer the green vector the greater the speed Before After Before After Ball 1 Ball 1 Ball 2 Ball 2 Box 1 Box 1 Box 2 Box 2 Box 1 Ball 1 Box 1 Ball 1 Box 2 Ball 2 Box 2 Ball 2 1 2 Physics 207 Lecture 12 Pg 7 Exercise Momentum Conservation Which box ends up moving fastest Examine the change in the momentum of the ball In the case of box 1 the balls momentum changes sign and so its net change is largest Since momentum is conserved the box must have the largest velocity to compensate A Box 1 1 B Box 2 C same 2 Physics 207 Lecture 12 Pg 8 A perfectly inelastic collision in 2 D Consider a collision in 2 D cars crashing at a slippery intersection no friction V v1 m1 m2 m1 m2 v2 before after If no external force momentum is conserved Momentum is a vector so px py and pz Physics 207 Lecture 12 Pg 9 A perfectly inelastic collision in 2 D If no external force momentum is conserved Momentum is a vector so px py and pz are conseved V v1 m1 m2 m1 m2 v2 before x dir px m1 v1 m1 m2 V cos y dir py m2 v2 m1 m2 V sin after Physics 207 Lecture 12 Pg 10 Elastic Collisions Elastic means that the objects do not stick There are many more possible outcomes but if no external force then momentum will always be conserved Start with a 1 D problem Before After Physics 207 Lecture 12 Pg 11 Billiards Consider the case where one ball is initially at rest after before pa pb vcm Pa F The final direction of the red ball will depend on where the balls hit Physics 207 Lecture 12 Pg 12 Billiards All that really matters is conservation momentum and energy Ch 10 11 Conservation of Momentum x dir Px m vbefore m vafter cos m Vafter cos y dir Py 0 m vafter sin m Vafter sin after before pafter pb F Pafter Physics 207 Lecture 12 Pg 13 Force and Impulse A variable force applied for a given time Gravity At small displacements a constant force t Springs often provide a linear force k x towards its equilibrium position Chapter 10 Collisions often involve a varying force F t 0 maximum 0 We can plot force vs time for a typical collision The impulse J of the force is a vector defined as the integral of the force during the time of the collision Physics 207 Lecture 12 Pg 14 Force and Impulse A variable force applied for a given time J reflects momentum transfer t t p J F dt dp dt dt dp F Impulse J area under this curve Transfer of momentum Impulse has units of Newton seconds ti t t tf Physics 207 Lecture 12 Pg 15 Force and Impulse Two different collisions can have the same impulse since J depends only on the momentum transfer NOT the nature of the collision F same area F t t big F small t t t t small F big Physics 207 Lecture 12 Pg 16 Average Force and Impulse F Fav F Fav t t big Fav small t t t t small Fav big Physics 207 Lecture 12 Pg 17 Exercise 2 Force Impulse Two boxes one heavier than the other are initially at rest on a horizontal frictionless surface The same constant force F acts on each one for exactly 1 second Which box has the most momentum after the force acts F A B C D light F heavy heavier lighter same can t tell Physics 207 Lecture 12 Pg 18 Boxing Use Momentum and Impuse to estimate g force Physics 207 Lecture 12 Pg 19 Back of the envelope calculation t J F dt Favg t 1 marm 7 kg 2 varm 7 m s 3 Impact time t 0 01 s Question Are these reasonable Impulse J p marm varm 49 kg m s F J t 4900 N 1 mhead 6 kg ahead F mhead 800 m s2 80 g Enough to cause unconsciousness 40 of fatal blow Only a rough estimate Physics 207 Lecture 12 Pg 20 Woodpeckers During collision with a tree ahead 600 1500 g How do they survive Jaw muscles act as shock absorbers Straight head trajectory reduces damaging rotations rotational motion is very problematic Physics 207 Lecture 12 Pg 21 Home Exercise The only force acting on a 2 0 kg object moving along the xaxis Notice that the plot is force vs time If the velocity vx is 2 0 m s at 0 sec what is vx at 4 0 s p m v Impulse m v J0 1 J1 2 J2 4 m v 8 1 N s 8 1 N s 16 2 N s m v 4 N s v 2 m s vx 2 2 m s 4 m s Physics 207 Lecture 12 Pg 22 Chapter 10 Energy We need to define an isolated system We need to define conservative force Recall chapter 9 force acting for a period of time gives an impulse or a change transfer of momentum What if a force acting over a distance Can we identify another useful quantity Physics 207 Lecture 12 Pg 23 …
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