Today s topic: IMPULSE AND MOMENTUM CONSERVATION

Transcription

1 Today s topic: IMPULSE ND MOMENTUM CONSERVTION

2 Reiew of Last Week s Lecture Elastic Potential Energy: x: displacement from equilibrium x = 0: equilibrium position Work-Energy Theorem: W tot W g W el W noncons K K K W noncons ( K U g U ) ( ) el K U g U el Let U U g U el E K U = Potential Energy = Total Mechanical Energy W non cons E E (friction, drag, work done by muscles, etc.) W non cons 0 E E Conseration of mechanical energy Force and Potential Energy: F x du dx du dx U(x) du dx x

3 Skate Park nimation

4 Slow motion ideo of last week s spring launch: What does the spring do, other than shooting up (and falling down)? Does ibrational/rotational motion store energy? What kind of energy? Did you account for this energy in last weeks workshop? Only about % of the total energy in ibration, much less in rotation!

5 Rotational State Populations Spring launch may sere as model for molecules desorbing (i.e., detaching) from a surface: F. M. Zimmermann and W. Ho, Surface Science Reports, 7-47 (995).

6 spring can ibrate in many normal modes : The higher the number of nodes, the greater the ibrational frequency. True not only for springs, but any solid! Do these modes continue to infinity (infinite # of nodes)? No, waelength is limited by spacing between atoms: Lattice Vibrations or Phonon Modes

7 Use femtosecond laser spectroscopy to measure phonon ibrations in LuMnO3 crystal: Reflectiity change x 0-5 Pump-Probe Delay (picoseconds) S. Lou, F. M. Zimmermann, R.. Bartynski, N. Hur, and S. Cheong, Physical Reiew B 79, 430 (00).

8 MOMENTUM & IMPULSE NEWTON S nd Law: Write differently: F Define Momentum: F ma d d m ( m) dt dt p m (Units: kg m/s = N s) dp F dt Net force = Rate of change of momentum Consider this relationship further: Define Impulse: t J t dp F dp ( F) dt dt p t dp F dt p p p ( F) dt t p p Vector that equals change in momentum

9 Had: Work-Energy Theorem, now hae: Impulse Momentum Theorem: Consider a ariable force acting on an object from time t to t (e.g., basketball dribble) J p p F(t) F t t t J t F( t) dt t J F ( t ) t Integral of actual force from to is equal to aerage force times interal t t t COMPRISON: MOMENTUM s. KINETIC ENERGY: p t is a ector ; KE is a scalar p t KE p KE related to time oer which force acts! related to distance oer which force acts!

10 i-clicker 0-kg box, initially at rest, moes along a frictionless horizontal surface. horizontal force to the right is applied to the box. The magnitude of the force changes as a function of time as shown.. The impulse in the first seconds is kg m/s B. The impulse from 5 seconds to 8 seconds is -6 kg m/s C. The impulse in the first seconds is kg m/s D. The impulse from seconds to 5 seconds is 0 kg m/s E. The impulse cannot be determines with the information gien

11 i-clicker E. I want 0 points subtracted from my grade J J p p 0 p

12 i-clicker -kg object accelerates in response to an applied force. During the 5-second interal that the force is applied, the object s elocity changes from 3 m/s east to 7 m/s west. Which is true about the magnitude of the impulse?. It equals 0 kg m/s B. It equals 8 kg m/s C. It equals 8/5 kg m/s D. It equals 4 kg m/s E. It cannot be found with the information gien.

13 i-clicker In Case, a metal bullet penetrates a wooden block. In Case B, a rubber bullet with the same initial speed and mass bounces off of an identical wooden block. Will the speed of the wooden block after the collision be greater in Case, greater in Case B, or the same in both cases?. The speed will be greater in Case because the metal bullet exerts a larger force on the block. B. The speed will be greater in Case B because the bullet changes direction. C. The speed will be the same in both cases because the bullets hae the same mass and initial speed and gie the block the same momentum. D. Cannot be determined.

14 CONSERVTION OF LINER MOMENTUM Consider two isolated objects that interact only by their mutual force. (No net external force) F B on F on B F F on B F B on B on (Newton s 3 rd Law) 0 B F on B But F B on dp dt F on B dp dt B So: dp dt dp dt B d dt ( p p ) B 0 For isolated system (no external forces) total linear momentum of the system is constant: P p p constant CONSERVTION OF LINER MOMENTUM B

15 Conseration of momentum is alid for any number of particles interacting only with each other (No External Forces) P i p i Is a ector quantity that is consered EXMPLE - physics of hockey: Ranger and a Deils hockey player are fighting on the ice. The Deils player (M = 00 kg) throws a punch that sends the Ranger (m = 80 kg) off at R 0.5 m/s. p D What is the speed of the Deils player,? R i 0; P f P p R p D p D P 0 i i P f 0 P p R f p i D 0 f 0 R f D f p R f ( 80 kg)( 0.5 m/s); p D f (00 kg) ( 40 kgm/s) (00 kg) D f (40 kg m/s) (00 kg) D 0.4 m/s f 0 D f

16 i-clicker Two boxes are tied together by a string and are sitting at rest in the middle of a large frictionless surface. Between the two boxes is a massless compressed spring. The string tying the two boxes together is cut suddenly and the spring expands, pushing the boxes apart. The box on the left has four times the mass of the box on the right. t the instant (after the string is cut) that the boxes lose contact with the spring, the speed of the box on the left will be.) B.) C.) D.) Greater than the right box Less than the right box Equal to the right box Not enough information proided

17 MOMENTUM CONSERVTION ND COLLISIONS Collision: Brief, strong interaction between objects. F ext F i If between objects, Neglect behaes as an isolated system P F i p i F i I Total momentum just after collision = Total momentum just before collision P I i p F ext Classify Collisions: Elastic Collision Total momentum and Total kinetic energy consered Inelastic Collision Momentum consered KE is not (lost to internal energy) Completely Inelastic Collision Momentum consered (Objects stick together) KE not. (KE Internal) Momentum consered in any collision KE consered only in elastic collision

18 EXMPLE: Completely Inelastic Collision The Ballistic Pendulum: h bullet is fired into clip of pendulum which swings to height h. What is? TWO PRTS! ( m a, ) Collision is completely inelastic Use P F P I to find state just after collision. P m 0 P ( m m I Use conseration of mechanical energy: K a U g K f U gf F a b) m m B m ( m mb ) ( m mb ) gh gh m mb m gh

19 Billiard Ball Collision -D Collision along x-axis (omit subscripts) P F P I ELSTIC COLLISION P KE and CONSERVED m Before fter m m m i m B B B B? B i m B KE F 3 KE I m m B B PGE OF LGEBR ( B ) ( ) B m m B B MGNITUDE OF RELTIVE VELOCITY UNCHNGED FTER COLLISION

20 EXMPLE: Pocket The Eight Ball Before collision: m moing m B at rest m m B 3 m m m 0 B B 0 B m m m B B 3 IMPORTNT CSES m m m m B B I mb m 0 ; B m m B II mb m B ; m m B III m m B ; B 0 m m B

21 i-clicker Carts and B are shown just before they collide. Which (if any) of the following statements could possibly be correct? I. fter the collision, the carts will stick together and moe off to the left due to Cart B haing more speed. II. They ll stick together and moe off to the right because Cart is heaier. III. The speed and the mass compensate. For completely inelastic collision, both carts are going to be at rest after the collision. IV. For an elastic collision, they will change their directions, so Cart will be moing to the left at 3 m/s and Cart B will be moing to the right at 4 m/s..) I B.) II C.) III D.) IV E.) III & IV

22 Collision in two dimensions (horizontal plane) Before: fter: Write separate momentum conseration equations for components: P P x y : mx mbbx mx mb Bx : my mbby my mb By If collision is elastic: KE KE Three equations, can sole for a maximum of three unknowns: Momentum and energy conseration alone are not sufficient to determine the final state.

23 i-clicker Two identical steel balls, S and T, are shown at the instant that they collide. The paths and elocities of the two balls before and after the collision are indicated by the dashed lines and arrows. What is the direction of the impulse on ball S?. B. Cannot be determined without the time t. C. C D. None of the other answers. E. E p i p f I = Δp

24 i-clicker Completely Inelastic Collision Momentum consered (Objects stick together) KE not. (KE Internal)