Physics 207 Lecture 12 Impulse Linear Momentum Physics 207 Lecture 12 Oct 15 Agenda Finish Chapter 9 start Chapter 10 Transition Chapter 9 Momentum Impulse Collisions Momentum conservation in 2D Impulse from forces to conservation laws Newton s Laws Conservation Laws Conservation Laws Newton s Laws Assignment HW5 HW6 They are different faces of the same physics phenomenon for special cases due Wednesday posted soon Physics 207 Lecture 12 Pg 1 Physics 207 Lecture 12 Pg 2 Lecture 12 Exercise 1 Momentum Conservation Lecture 12 Exercise 1 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 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 In which case does the box ends up moving fastest No external force then vectorially COM A B C A Box 1 Box 1 Box 2 Same C same 2 1 2 1 B Box 2 Physics 207 Lecture 12 Pg 3 Physics 207 Lecture 12 Pg 4 Elastic Collisions A perfectly inelastic collision in 22 D Consider a collision in 2 D cars crashing at a slippery intersection no friction m1 m2 m1 m2 There are many more possible outcomes but if no external force then momentum will always be conserved Start with a 1 D problem v2 before Elastic means that the objects do not stick V v1 after If no external force momentum is conserved Momentum is a vector so px py and pz Before Physics 207 Lecture 12 Pg 5 After Physics 207 Lecture 12 Pg 6 Page 1 Physics 207 Lecture 12 Force and Impulse A variable force applied for a given time Elastic Collision in 11 D Gravity usually a constant force to an object Springs often provide a linear force k x towards its equilibrium position Chapter 10 m2 m1 before v2b v1b x 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 m2 m1 after v2a v1a Physics 207 Lecture 12 Pg 7 Physics 207 Lecture 12 Pg 8 Force and Impulse A variable force applied for a given time Force and Impulse J reflects momentum transfer Two different collisions can have the same impulse since J depends only on the momentum transfer NOT the nature of the collision r r J t F dt t dpr dt dt p dpr F F Impulse J area under this curve Transfer of momentum ti t t t Impulse has units of Newton seconds same area F t t big F small tf Physics 207 Lecture 12 Pg 9 t t t small F big Physics 207 Lecture 12 Pg 10 Average Force and Impulse Example from last time A 2 kg cart initially at rest on frictionless horizontal surface is acted on by a 10 N horizontal force along the positive x axis for 2 seconds what is the final velocity F Fav F is in the x direction F ma so a F m 5 m s2 F Fav t t big Fav small t t t v v0 a t 0 m s 2 x 5 m s 10 m s x direction but mv F t which is the area with respect to F t curve t small Fav big Physics 207 Lecture 12 Pg 11 Physics 207 Lecture 12 Pg 12 Page 2 Physics 207 Lecture 12 Lecture 12 Exercise 2 Force Impulse Lecture 12 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 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 Which box has the most momentum after the force acts A heavier F A B C D F light B lighter C same heavy heavier lighter same can t tell F F light heavy Physics 207 Lecture 12 Pg 13 Physics 207 Lecture 12 Pg 14 Back of the envelope calculation Boxers r r r J t F dt Favg t 2 varm 7 m s 3 Impact time t 0 01 s 1 marm 7 kg 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 Physics 207 Lecture 12 Pg 15 Enough to cause unconsciousness 40 of fatal blow Physics 207 Lecture 12 Pg 16 Woodpeckers Chapter 10 Energy During collision with a tree ahead 600 1500 g What do we mean by an isolated system What do we mean by a conservative force If a force acting on an object act for a period of time then we have an Impulse change transfer of momentum What if we consider this force acting over a distance Can we identify another useful quantity 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 17 Physics 207 Lecture 12 Pg 18 Page 3 Physics 207 Lecture 12 Energy Energy mg y m vy2 vy02 Fy m ay and let the force be constant t2 y t2 y t y0 vy0 t ay y t y0 vy0 t ay vy t vy0 ay t t vy vy0 ay vy ay So y vy0 vy ay ay vy ay 2 vy vy0 vy02 ay vy2 2vy vy0 vy02 ay mg yf yi m vyf2 vyi2 A relationship between y displacement and y speed Rearranging 2 ay y vy2 vy02 m vyi2 mgyi m vyf2 mgyf may y m vy2 vy02 Finally We associate mgy with the gravitational potential energy mg y m vy2 vy02 If falling Physics 207 Lecture 12 Pg 19 Physics 207 Lecture 12 Pg 20 Energy Energy Notice that if we only consider gravity as the external force then then the x and z velocities remain constant To If only conservative conservative forces are present the total energy sum of potential U and kinetic energies K K of a system is conserved m vyi2 mgyi m vyf2 mgyf Add m vxi2 m vzi2 and m vxf2 m Emech K U vzf2 Emech K U constant m vi2 mgyi m vf2 mgyf vi2 vxi2 vyi2 vzi2 where K and U may change but E K Umech remains a fixed value Emech is called mechanical energy m v2 terms are referred to as kinetic energy Physics 207 Lecture 12 Pg 21 Physics 207 Lecture 12 Pg 22 Another example of a conservative system …
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