Lecture 17. Conservation of Linear Momentum

Transcription

1 Lecture 17 Chapter 11 Physics I Conservation of Linear Momentum Course website: Department of Physics and pplied Physics

2 IN THIS CHPTER, you will learn to use the concepts of linear momentum. Today we are going to discuss: Chapter 11: Conservation of Linear Momentum: Section D Collisions (Elastic/Inelastic): Section 11.3 Department of Physics and pplied Physics

3 F d p dt t f p Fdt dp f t i p i Impulse During the collision, objects are deformed because of the large forces involved. How to relate those forces with a change in momentum? From Newton s 2 nd law: Integrate it: Fdt p f p i dp p p Impulse= change in momentum Define Impulse as: t t f i Fdt efore collision p i fter collision Force exerted on the ball p f Recall: Integral Graphical meaning is an area So, Impulse= area under F-vs-t curve Work-KE Principle Similarity Momentum Principle p Impulse t f t i Fdt Department of Physics and pplied Physics

4 Example Tennis all/racket Collision Force (kn) The force exerted by a tennis racket on the ball (mass 56 g) during a serve ( v i 0 ) can be approximated by the F vs time plot below. What is the impulse on the ball? What is the speed of the serve? 0 2 C Time (ms) 10 Momentum Principle Department of Physics and pplied Physics rea under force-time curve is an impulse, : rea p 2 rea( C) v f t t f i Fdt p f mv f v f m p f p i 2kN 2ms 2( ) 2 4 N s 71m/s kg 4N s

5 ConcepTest 2.0 kg object moving to the right with speed 0.50 m/s experiences the force shown. What are the object s speed and direction after the force ends? v i =0.5m/s x Kicking all ) 0.5 m/s left ) t rest C) 0.5 m/s right D) 1.0 m/s right v f -? x E) 2.0 m/s right Δp = p f - p i = p f = p i +.. /

6 Impulse/verage force The exact variation of F with time is very often not known (too complicated). How to get at least something? It is easier to find an average force. Let s keep the same impulse nd the same time interval Δt ut instead of a real force we will use an average force t t f i Fdt F avg F avg t t p p Department of Physics and pplied Physics

7 Example How to avoid broken legs for a cat? F avg t p Initial linear momentum ( ) F, that is what can break cat s bones and the cat feels that and tries to reduce F as much as it can. Since the cat falls from a certain height, Δp is given and the cat cannot do anything about that during the fall. y bending legs and increasing an impact time, Δt. Having a certain p, a cat by bending its lags tries to increase t (impact time), so that an impact force would be reduced. (intuitive knowledge of Physics ) Department of Physics and pplied Physics Final linear momentum

8 Example verage Force on a baseball The speed of a fastball is about 40 m/s, and the speed of the ball coming off of player s bat for a home run is about 54 m/s. The ball (0.145kg) is in contact with the bat for 1ms. What is the average Force exerted by the player? x F average F avg t p F average p t p f pi m( v t f ( v )) t Pay attention to directions!!!!!!!! i ( 0.145kg)[54 40] m / s F average s F average 136, 300N in the direction of v f or x Department of Physics and pplied Physics

9 Conservation of Linear Momentum Department of Physics and pplied Physics

10 Conservation of linear momentum (derivation) Let s consider some cruelty Consider two colliding particles (a ball and a bird). ssume: there are external and internal forces Internal forces obey N.3 rd law: F F From modified Newton s 2 nd law during the collision: dp ext F F dt dd these two eq-ns dp ext ext F F F dt F d( p p) ext ext F F F ( F) dt Thus, the internal forces cancel each other!!! Let s introduce a total linear momentum of two objects Thus, we don t need to know the internal forces to solve collision problems. Department of Physics and pplied Physics ext ext F F dp dt P F F p p ext ext F ext F Cont. collision ext F F

11 Conservation of Linear Momentum Let s look at a special case of zero net external forces. If F ext 0 dp F dt dp, then 0, dt P const thus If no net external force acts on a system, its momentum is conserved. gain, the total momentum remains constant, regardless of whatever interactions (internal forces) are going on inside the system ext Department of Physics and pplied Physics

12 Why is momentum conserved during collision? F N1 F N 2 m g m g 1 2 mg s are canceled by normal forces, so net external force is zero, and the momentum is conserved m g 1 F N 2 m g 2 The net external force is m 1 g, and the momentum is NOT conserved Isolated system is a system on which no external forces act. There are only internal forces acting between objects. this system (two balls) is isolated this system (two balls) is NOT isolated The total momentum of an isolated system of objects is conserved Department of Physics and pplied Physics

13 Example Rolling away girl running with speed of 4.0 m/s jumps on a stationary cart. The girl has a mass of 30 kg and the cart s mass is 10 kg. What is the cart s speed just after the girl jumps on? m 30kg v v 0 m 10kg v v? Initial (before collision) pply Conservation of Momentum: P initial P final Final (after collision) (Since this system is isolated) p initial mv m P initial P final v m v m v p final ' ( m m ) v m v ' m v ' ' ( m m ) v Solve for v : v mv m m 30kg 4.0m / s 30kg 10kg 3m / s Department of Physics and pplied Physics

14 Different types of collisions Linear momentum is conserved in both of these two-body collisions (since there is no net external force) Is mechanical energy conserved in these collisions? Metal balls get deformed and restored Mech. energy is conserved Elastic collision Cars get deformed and not restored. Some Mech. Energy is spent on deformation. Mech. energy is NOT conserved Inelastic collision Department of Physics and pplied Physics

15 1-D Elastic Collisions Department of Physics and pplied Physics

16 m v Elastic Collision Math (1D) m v m v m v Conservation of momentum Conservation of mech. energy m v m v m v m v 1 2 m v 2 1 m 2 v 2 1 m v m 2 2 m v m v m v m v m v 2 m v 2 m v 2 2 m v m (v v ) m ( v v ) Relative velocities switch signs in the collision Department of Physics and pplied Physics v v m (v 2 v 2 ) m ( v 2 v 2 ) m (v v )(v v ) m ( v v )( v v ) ( v v ) v v v v v

17 Elastic Collision Math (1D) 1 st m v m v m v m v 1 m v m v m v m 2 v 2 Conservation of momentum Conservation of mechanical energy 2 nd m v m v m v m v v ( v v ) v Conservation of momentum Conservation of mechanical energy So, instead of the 1 st set of crazy equations, we can use the 2 nd one which is easier (both are linear). It is only true for an elastic head-on collision Department of Physics and pplied Physics

18 Example allistic Pendulum device used to measure the speed of a bullet. the speed of a bullet Department of Physics and pplied Physics For ullet mass 10 g lock mass is 3 kg lock swings up to a height of 5 cm v o 298 m s

19 Department of Physics and pplied Physics Thank you See you on Monday Exam 3