Phs0 Lectures 3, 4 Moentu and ollisions Ke points: Moentu and ipulse ondition for conservation of oentu and wh How to solve collision probles entre of ass Ref: 7-,,3,4,5,6,7,8,9,0. Page
Moentu is a vector: It s a quantit that represents the aount and direction of otion. It was once called the quantit of otion. Now we know that kinetic energ is another quantit of otion. Newton s nd Law dv dp p F a dt dt t p F t J where J F t is called the ipulse of force. Ipulse-oentu principle : The net ipulse on an object is equal to the change in oentu of the object. J p F
Eaple: Force of a tennis serve. For a top plaer, a tennis ball a leave the racket on the serve with a speed of 55 /s (about 0 i/h). If the ball has a ass of 0.060 kg and is in contact with the racket for about 4 s (4 0-3 s), estiate the average force on the ball. [Solution] J F ave t p p v 0 Fave t t 0.0655 0.004 800 ( N)
i-clicker question 3-0.40-kg baseball is dropped fro rest. It has a speed of.0 /s just before it hits the ground. It rebounds with a speed of.00 /s. The ball is in contact with the ground for 0.040 s. What is the average force eerted b the ground on the ball during that tie? ).00 N ) 0.00 N ).0 N D) 4.0 N E).0 N F t F F t p p p ( v f v ) i t t 0. 40. 00 (. 0 ) 0. 04 ( N )
Man-bod Sste Two students pulling each other on ice. Internal forces: and. Eternal forces: Newton s law: F g, g F N F g, FN and F N F dding the two equations g FN g F dp Fet Fint dt dp dt Since the internal forces alwas cancel, dp F et (onl eternal forces can change dt N. F P, F g dp dt F F N dp dt F F dp dt is the total oentu of F the sste F, F 0 int the total oentu) Deo
onservation of Moentu When F et 0, dp dt 0, i.e., P cons tant This is the law of conservation of linear oentu: When the net eternal force on a sste of objects is zero, the total oentu of the sste reains constant. Note : If one of the coponents of the net eternal force is zero, the corresponding coponent of the total oentu of the sste is conserved (even though the total oentu vector a or a not be conserved). Note : For a one-object sste, the condition for oentu conservation is that the net force acting on the object is zero.
i-clicker question 3- The condition for oetu to be conserved is () It s a closed sste. () The net eternal force is zero. () No nonconservative work. (D) The oentu is never conserved. (E) The oentu is alwas conserved.
Eaple: Rifle recoil. alculate the recoil velocit of a 5.0-kg rifle that shoots a 0.00-kg bullet at a speed of 60 /s. [Solution] X-coponent of eternal Force is zero; Thus the -coponent of the total oentu is conserved: P f v R v R v P i. 5 R 0 v v R R ( / s ) 0 0. 060 5. 5 ( / s )
ollisions Moentu is conserved in all collisions. Wh? ecause the ipulse of eternal forces can be ignored (uch saller than internal ipulse). ollisions in which kinetic energ is conserved as well are called elastic collisions, and those in which it is not are called inelastic.
Elastic ollisions in One Diension Here we have two objects colliding elasticall. We know the asses and the initial speeds. Since both oentu and kinetic energ are conserved, we can write two equations. This allows us to solve for the two unknown final speeds.
Eaple: illiard ball of ass oving with speed v collides head-on with ball of equal ass. What are the speeds of the two balls after the collision, assuing it is elastic? ssue ball is initiall at rest (v = 0). onservation of oentu: v v v ( - cop) onservation of kinetic energ: v v v v v v v v v () () () () : 0 vv Solution : Solution : v 0 v v v v 0 Solution should be rejected because it eans no collision. v
Inelastic ollisions With inelastic collisions, soe of the initial kinetic energ is lost to theral or potential energ. Kinetic energ a also be gained during eplosions, as there is the addition of cheical or nuclear energ. copletel inelastic collision is one in which the objects stick together afterward, so there is onl one final velocit. v v
Eaple: Railroad cars collide 0,000-kg railroad car,, traveling at a speed of 4.0 /s strikes an identical car,, at rest. If the cars lock together as a result of the collision, what is their coon speed iediatel after the collision? onservation of oentu (-cop.): v v v Hit and stick (perfectl inelastic): v v Solve: v v v
Eaple: allistic pendulu. The ballistic pendulu is a device used to easure the speed of a projectile, such as a bullet. The projectile, of ass, is fired into a large block of ass M, which is suspended like a pendulu. s a result of the collision, the pendulu and projectile together swing up to a aiu height h. Deterine the relationship between the initial horizontal speed of the projectile, v, and the aiu height h. [Solution] Two events:. Hit and stick;. Swing. v ( M )v ( M )v ( M )gh Eliinate M v: v ( M )gh, v gh M
ollisions in Two or Three Diensions Eaple: Proton-proton collision. proton traveling with speed 8. 0 5 /s collides elasticall with a stationar proton in a hdrogen target. One of the protons is observed to be scattered at a 60 angle. t what angle will the second proton be observed, and what will be the velocities of the two protons after the collision? Elastic collision: v i.e., v v v v v () onservation of oentu: v : v v : 0 v v v sin cos, i.e., v v v cos sin v v () (3), 3 unknowns : v,v,. 3 equations : (), (), (3).
i-clicker question 3-3 car with a ass of 00 kg and a speed of /s heading north approaches a intersection. t the sae tie, a inivan with a ass of 300 kg and speed of 4 /s heading east is also approaching the intersection. The car and the inivan collide and stick together. onsider the total oentu and the total kinetic energ of the two vehicles before and after the collision. What. is the oth velocit the total of oentu the wrecked and vehicles total kinetic just after energ the collision? are conserved. Hit and stick: v v V vr. The total oentu is conserved but the total kinetic (N) onservation energ of oentu: is not conserved. p pv P R. Neither the total oentu nor the total kinetic energ is conserved. P R V vv v 34400 ( kg / s ) D. The total kinetic energ is conserved but the total (E) oentu P R is 34400 not conserved. vr 3. 7 ( / s ) E. The change in total oentu equals the change in total V 300 00 p V kinetic energ. tan p p V tan V v v V 65 p P R
enter of Mass (M) In (a), the diver s otion is pure translation; in (b) it is translation plus rotation. There is one point that oves in the sae path a particle would take if subjected to the sae force as the diver. This point is called the center of ass (M).
enter of Mass (M) The general otion of an object can be considered as the su of the translational otion of the M, plus rotational, vibrational, or other fors of otion about the M.
enter of Mass (M) For two particles, the center of ass lies closer to the one with the ost ass: where M is the total ass. In general,,, M M
Eercise: Three particles in -D. Three particles, each of ass.50 kg, are located at the corners of a right triangle whose sides are.00 and.50 long, as shown. Locate the center of ass. [Solution] ) (... M 33 0 0 0 3 ) ( 3 ( ).. M 0 50 5 0 0 3 3
Eaple: M of L-shaped flat object. Deterine the M of the unifor thin L-shaped object shown. [Solution] The object consists of two rectangular parts: and, whose centres of ass are (, ) and (, ).. 03,. 96, t(. 060. 0 ) 0. 4t 0. 0 0. 74 ( densit, t(. 48 0. 0 ) 0. 96t t thickness ) M M 0. 4. 03 0. 96. 96. 4 ( ) 0. 4 0. 96 0. 40. 0 0. 96( 0. 74 ) 0. 5 ( ) 0. 4 0. 96
enter of Mass and Translational Motion The total oentu of a sste of particles is equal to the product of the total ass and the velocit of the center of ass. P M Total v The su of all the forces acting on a sste is equal to the total ass of the sste ultiplied b the acceleration of the center of ass: Therefore, the center of ass of a sste of particles (or objects) with total ass M oves like a single particle of ass M acted upon b the sae net eternal force.
onceptual Eaple: two-stage rocket. rocket is shot into the air as shown. t the oent it reaches its highest point, a horizontal distance d fro its starting point, a prearranged eplosion separates it into two parts of equal ass. Part I is stopped in idair b the eplosion and falls verticall to Earth. Where does part II land? ssue g = constant.