Today s s topics are: Collisions and Momentum Conservation. Momentum Conservation

Transcription

1 Today s s topics are: Collisions and P (&E) Conservation Ipulsive Force Energy Conservation How can we treat such an ipulsive force? Energy Conservation

2 Ipulsive Force and Ipulse [Exaple] an ipulsive force on a baseball that is struck with a bat has: <F> ~ 5000 N & t ~ 0.0 s The ipulse concept is ost useful for ipulsive forces. Ipulsive Force Very short tie Very large agnitude Work vs. Ipulse Work Ipulse Distance, x K (/) Work-Energy Theore Energy Conservation p Ipulse-Moentu Theore

3 Ipulse-Moentu Theore F (t) D J < F > D dv d( v) dp F a dt dt dt F dt dp J t t f i F dt p f p i () dp p f p i p p f < F > t t f < F > ( t t ) f i p f i ti p p i Exaple f 8 /s What is the ipulse given to the wall? Note: kg. i 8 /s 3

4 F x ( t) J x p x, f p x,i y x F x ( t) F x ( t) J x J x ( ) F x t J x p x, f p x,i p x,i + p x,i p x, f + p x, f Energy Conservation K +,i + K,i K, f K, f K,i + K,i K, f + K, f + Q Loss of energy as theral and other fors of energy 4

5 D D Collision M M D D Collision 5

6 D/D Explosion (or ore) Exaple Before collision After collision (totally inelastic collision) 6

7 Exaple 3 Express and in ters of, M, g, and h. (A) What Is Total Inelastic Collision? When two objects collide and stick together after a collision, the axiu possible fraction (fraction < 00%) of the initial kinetic energy is transfored by conserving the oentu of the syste of two objects. This collision is called totally (or perfectly) inelastic. This axiu possible fraction does not necessary ean K f 0 (fraction 00%) in a case where p i(syste) is not zero. The figure in the left is such an exaple. Q: Can you specify a type of collision where 00% of the initial kinetic energy is transfored? (B) Energy Conservation 7

8 Exaple 4 Proton-proton elastic collision: kg /s θ 60.0 o? θ? Proble 5 A 000-kg two stage rocket is traveling at a speed of /s away fro the Earth when a pre-designed explosion separates the rocket into two sections of 00 kg and 900 kg. (We assue that a loss of ass due to the explosion is negligible.) The 900-kg section oves in a direction perpendicular to the original line of otion with a speed of /s. Ignore any gravitational forces fro the Earth and other planets. (a) What is the speed and direction of the 00-kg section (relative to the original line of otion) after the explosion? (b) How uch energy was supplied by the explosion? 8

9 Defining Syste and F syste Syste Before collision After collision (totally inelastic collision) y x Exaple Center of Mass (CM) Exaple Center of Mass (c.. or CM) The overall otion of a echanical syste can be described in ters of a special point () called center of ass of the syste: Fsyste M syste ac where F is the vector su of all the forces syste exerted on the syste. 9

10 0 CM Kineatics CM Kineatics ( Body Body Syste) Syste) a a a v v v r r r c c c ; M syste p syste Dynaics Dynaics (general) (general) constant 0 d d d d d d syste syste syste c syste c syste c syste syste p F t p t v M t v M a M F ) ( Conservation of oentu

11 CM Position and Velocity syste 0 F syste e.g., Totally inelastic collision: + ( + ) r + r v + v rc ; vc t s x t 0 s c c /s /s Wearing golf shoes Exaple 6: F syste 0 50 kg kg Frictionless surface x 40 kg d

12 Question: When should we use law of energy conservation or/and law of oentu conservation? Step : Defining Syste and F syste Syste Before collision After collision (totally inelastic collision) y x

13 Step : Reeber work vs. ipulse Change of the status (K) of syste over position Change of the status (p) of syste over tie Work Ipulse Distance, x K (/) Work-Energy Theore Energy Conservation p Ipulse-Moentu Theore Glossary. K: Energy associated with the otion of an object.. U: Energy stored in a syste of objects Can either do work or be converted to K. 3. Q: Theral Energy (Internal Energy) The energy of atos and olecules that ake up a body. 3

14 Step 3: Analysis of E and P Conservation B Skeet+Pellet A C Exaple: Ballistic Pendulu B Skeet+Pellet (A) A (B) Energy Conservation C 4

15 Exaple: Ballistic Pendulu B Skeet+Pellet (A) A (B) Energy Conservation P.8-87 C 9-54, elastic collision Final velocity of? Proble 7 Proble : (5 points) Two billiard balls of asses ( 4M) and ( M) ove at right angles and eet at the origin of an x-y coordinate syste. One is oving upward along the y axis at.00 /s, and the other is oving to the right along the x axis with speed 3.70 /s. After the collision (assued elastic), the second ball is oving along the positive y axis. What is the final direction of the first ball? What are their two speeds? 5

16 3 CM Position (D) CM Position (D) 3 y c x c

17 Path of C.M. ( ) in is stopped and falls vertically I 0 /s M syste I II M syste F Analysis of C.M. ( ) in syste M syste (0) iˆ + ( g) ˆj a c x direction : Motion w/ constant velocity y direction : Motion w/ constant acceleration 7

18 Proble 8 A 0.0-kg projectile is fired at an angle of 60.0 o above the horizontal and with a speed of 40 /s. At the highest point of its trajectory explodes into two fragents, one of which has a ass of 5.0 kg and falls vertically with an initial speed of 0.0 /s. Neglect air resistance. (a) How far fro the point of firing does the other fragent strikes if the terrain is level? (b) How uch energy is released during the explosion? (c) Find the position (x and y) of the center of ass syste of two fragents at tie of 0 seconds after the explosion. Proble 9 A 0.50-kg projectile is fired at an angle of 30.0 o above the horizontal and with a speed of 30.0 /s. When it reaches the axiu height, it is hit fro below by a kg bullet traveling vertically upward at a speed of 00 /s. The bullet is ebedded in the projectile. Neglect air resistance. (a) Calculate the axiu height just before the collision. (b) Calculate the velocity of the projectile at the axiu height right after collision. (c) How uch energy is given or lost by the collision? (d) How uch higher (after the collision) did the projectile go up? (e) Calculate how far away the projectile hits the ground. 8

19 Suary:. Syste C.M.. Kineatics and Dynaics 3. Analysis of Motion of C.M. 4. Collision and Explosion 5. (Don t t ignore ipulse!) ipulse!) 9