Momentum and Collisions. Rosendo Physics 12-B

Transcription

1 Moentu and Collsons Rosendo Physcs -B

2 Conseraton o Energy Moentu Ipulse Conseraton o Moentu -D Collsons -D Collsons The Center o Mass Lnear Moentu and Collsons February 7, 08

3 Conseraton o Energy D E = D K + D U = 0 conserate orces are the only orces that do work on the syste. The total aount o energy n the syste s constant. gy kx D E = D K + D U = - k d rcton orces are dong work on the syste. The total aount o energy n the syste s stll constant, but the change n echancal energy goes nto nternal energy or heat. k d gy kx gy kx gy kx February 7, 08

4 Lnear Moentu Ths s a new undaental quantty, lke orce, energy. It s a ector quantty (ponts n sae drecton as elocty). The lnear oentu p o an object o ass ong wth a elocty s dened to be the product o the ass and elocty: p The ters oentu and lnear oentu wll be used nterchangeably n the text Moentu depend on an object s ass and elocty February 7, 08

5 Moentu and Energy Two objects wth asses and hae equal knetc energy. How do the agntudes o ther oenta copare? (A) p < p (B) p = p (C) p > p (D) Not enough noraton s gen February 7, 08

6 Lnear Moentu, cont d Lnear oentu s a ector quantty Its drecton s the sae as the drecton o the elocty The densons o oentu are ML/T The SI unts o oentu are kg / s Moentu can be expressed n coponent or: p x = x p y = y p z = z p February 7, 08

7 Newton s Law and Moentu Newton s Second Law can be used to relate the oentu o an object to the resultant orce actng on t D D( ) F net a Dt Dt The change n an object s oentu dded by the elapsed te equals the constant net orce actng on the object Dp Dt changen oentu te nteral F net February 7, 08

8 Ipulse When a sngle, constant orce acts on the object, there s an pulse delered to the object I F Dt s dened as the pulse I The equalty s true een the orce s not constant Vector quantty, the drecton s the sae as the drecton o the orce Dp changen oentu F net Dt te nteral February 7, 08

9 Ipulse-Moentu Theore The theore states that the pulse actng on a syste s equal to the change n oentu o the syste I Dp Dp F net Dt I February 7, 08

10 Calculatng the Change o Moentu Dp p p ater beore ater beore ( ) ater beore For the teddy bear Dp 0 ( ) For the bouncng ball Dp ( ) February 7, 08

11 How Good Are the Bupers? In a crash test, a car o ass kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are =-5 /s and =.6 /s, respectely. I the collson lasts or 0.5 s, nd (a) the pulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car February 7, 08

12 How Good Are the Bupers? In a crash test, a car o ass kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are =-5 /s and =.6 /s, respectely. I the collson lasts or 0.5 s, nd (a) the pulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car p ( kg)( 5 / s) kg / s p ( kg)(.6 / s) kg / s I Dp Dt p I Dt p ( kg / s) (.50 kg / s kg / s 0.5s kg / s) 5 F a N February 7, 08

13 Ipulse-Moentu Theore A chld bounces a 00 g superball on the sdewalk. The elocty o the superball changes ro 0 /s downward to 0 /s upward. I the contact te wth the sdewalk s 0.s, what s the agntude o the pulse parted to the superball? (A) 0 (B) kg-/s (C) 0 kg-/s (D) 00 kg-/s (E) 000 kg-/s I Dp February 7, 08

14 Ipulse-Moentu Theore A chld bounces a 00 g superball on the sdewalk. The elocty o the superball changes ro 0 /s downward to 0 /s upward. I the contact te wth the sdewalk s 0.s, what s the agntude o the orce between the sdewalk and the superball? (A) 0 (B) N I Dp F (C) 0 N Dt Dt Dt (D) 00 N (E) 000 N February 7, 08

15 Conseraton o Moentu In an solated and closed syste, the total oentu o the syste reans constant n te. Isolated syste: no external orces Closed syste: no ass enters or leaes The lnear oentu o each colldng body ay change The total oentu P o the syste cannot change. February 7, 08

16 Conseraton o Moentu Start ro pulse-oentu theore FDt FDt Snce Then So F t F D Dt ( ) February 7, 08

17 Conseraton o Moentu When no external orces act on a syste consstng o two objects that collde wth each other, the total oentu o the syste reans constant n te F Dt Dp p p net When F net 0 then For an solated syste p p Dp 0 Speccally, the total oentu beore the collson wll equal the total oentu ater the collson February 7, 08

18 The Archer An archer stands at rest on rctonless ce and res a 0.5-kg arrow horzontally at 50.0 /s. The cobned ass o the archer and bow s 60.0 kg. Wth what elocty does the archer oe across the ce ater rng the arrow? p p 60.0kg, 0.5kg, 0, 50 / s,? 0 0.5kg (50.0 / s) 0.47 / 60.0kg s February 7, 08

19 Conseraton o Moentu A 00 kg an and 50 kg woan on ce skates stand acng each other. I the woan pushes the an backwards so that hs nal speed s /s, at what speed does she recol? (A) 0 (B) 0.5 /s (C) /s (D).44 /s (E) /s February 7, 08

20 Types o Collsons Moentu s consered n any collson Inelastc collsons: rubber ball and hard ball Knetc energy s not consered Perectly nelastc collsons occur when the objects stck together Elastc collsons: bllard ball both oentu and knetc energy are consered Actual collsons Most collsons all between elastc and perectly nelastc collsons February 7, 08

21 Collsons Suary In an elastc collson, both oentu and knetc energy are consered In a non-perect nelastc collson, oentu s consered but knetc energy s not. Moreoer, the objects do not stck together In a perectly nelastc collson, oentu s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the sae Elastc and perectly nelastc collsons are ltng cases, ost actual collsons all n between these two types Moentu s consered n all collsons February 7, 08

22 More about Perectly Inelastc Collsons When two objects stck together ater the collson, they hae undergone a perectly nelastc collson Conseraton o oentu ) ( Knetc energy s NOT consered February 7, 08

23 An SUV Versus a Copact An SUV wth ass kg s traellng eastbound at +5.0 /s, whle a copact car wth ass kg s traellng westbound at -5.0 /s. The cars collde head-on, becong entangled. (a) Fnd the speed o the entangled cars ater the collson. (b) Fnd the change n the elocty o each car. (c) Fnd the change n the knetc energy o the syste consstng o both cars. February 7, 08

24 An SUV Versus a Copact (a) Fnd the speed o the entangled cars ater the collson. p p ) ( kg, kg, 5 / s 5 / s 5.00 / s February 7, 08

25 An SUV Versus a Copact (b) Fnd the change n the elocty o each car. D D 5.00 / s 0.0 / 0.0 / s s kg, kg, 5 / s 5 / s 4 D ( ).80 kg / s 4 D ( ).80 kg / s D D 0 February 7, 08

26 An SUV Versus a Copact (c) Fnd the change n the knetc energy o the syste consstng o both cars / s kg, kg, 5 / s 5 / s KE KE J J DKE KE KE J February 7, 08

27 More About Elastc Collsons Both oentu and knetc energy are consered Typcally hae two unknowns Moentu s a ector quantty Drecton s portant Be sure to hae the correct sgns Sole the equatons sultaneously February 7, 08

28 February 7, 08 Elastc Collsons A spler equaton can be used n place o the KE equaton ) ( ) )( ( ) )( ( ) ( ) ( ) ( ) (

29 Suary o Types o Collsons In an elastc collson, both oentu and knetc energy are consered In an nelastc collson, oentu s consered but knetc energy s not In a perectly nelastc collson, oentu s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the sae ) ( February 7, 08

30 Conseraton o Moentu An object o ass oes to the rght wth a speed. It colldes head-on wth an object o ass 3 ong wth speed /3 n the opposte drecton. I the two objects stck together, what s the speed o the cobned object, o ass 4, ater the collson? (A) 0 (B) / (C) (D) (E) 4 February 7, 08

31 Proble Solng or D Collsons, Coordnates: Set up a coordnate axs and dene the eloctes wth respect to ths axs It s conenent to ake your axs concde wth one o the ntal eloctes Dagra: In your sketch, draw all the elocty ectors and label the eloctes and the asses February 7, 08

32 Proble Solng or D Collsons, Conseraton o Moentu: Wrte a general expresson or the total oentu o the syste beore and ater the collson Equate the two total oentu expressons Fll n the known alues February 7, 08

33 Proble Solng or D Collsons, 3 Conseraton o Energy: I the collson s elastc, wrte a second equaton or conseraton o KE, or the alternate equaton Ths only apples to perectly elastc collsons Sole: the resultng equatons sultaneously February 7, 08

34 One-Denson s Two-Denson February 7, 08

35 Two-Densonal Collsons For a general collson o two objects n twodensonal space, the conseraton o oentu prncple ples that the total oentu o the syste n each drecton s consered x x x x y y y y February 7, 08

36 Two-Densonal Collsons The oentu s consered n all drectons Use subscrpts or Identyng the object Indcatng ntal or nal alues The elocty coponents I the collson s elastc, use conseraton o knetc energy as a second equaton Reeber, the spler equaton can only be used or one-densonal stuatons x y x y x y x y February 7, 08

37 Glancng Collsons The ater eloctes hae x and y coponents Moentu s consered n the x drecton and n the y drecton Apply conseraton o oentu separately to each drecton x y x y x y x y February 7, 08

38 -D Collson, exaple Partcle s ong at elocty and partcle s at rest In the x-drecton, the ntal oentu s In the y-drecton, the ntal oentu s 0 February 7, 08

39 -D Collson, exaple cont Ater the collson, the oentu n the x-drecton s cos q cos Ater the collson, the oentu n the y-drecton s sn q sn cosq sn q sn cos I the collson s elastc, apply the knetc energy equaton February 7, 08

40 Collson at an Intersecton A car wth ass kg traelng east at a speed o 5 /s colldes at an ntersecton wth a kg an traelng north at a speed o 0 /s. Fnd the agntude and drecton o the elocty o the wreckage ater the collson, assung that the ehcles undergo a perectly nelastc collson and assung that rcton between the ehcles and the road can be neglected. c cx / s, kg, y.50 0 / s, 3 kg? q? February 7, 08

41 p p x x c cx.50 Collson at an Intersecton 3 5/s, c c p p y y 4 c c cx cx kg, y x x.50 0 /s, c cx 3 kg? q? ( c ) kg/s ( cy cy y y y 3 kg) cosq 4 cosq ( c ) kg/s ( kg) sn q 4 snq kg/s kg /s February 7, 08

42 c cx.50 Collson at an Intersecton 3 5 / s, kg, y.50 0 / s, 3 kg? q? kg/s ( kg/s ( kg) kg) snq cosq kg / s tanq kg / s.33 q tan (.33) kg/s 3 (4.000 kg)sn /s February 7, 08

43 The Center o Mass How should we dene the poston o the ong body? What s y or U g = gy? Take the aerage poston o ass. Call Center o Mass (COM or CM) February 7, 08

44 The Center o Mass There s a specal pont n a syste or object, called the center o ass, that oes as all o the ass o the syste s concentrated at that pont The CM o an object or a syste s the pont, where the object or the syste can be balanced n the unor gratatonal eld February 7, 08

45 The Center o Mass The center o ass o any syetrc object les on an axs o syetry and on any plane o syetry I the object has unor densty The CM ay resde nsde the body, or outsde the body February 7, 08

46 Where s the Center o Mass? The center o ass o partcles Two bodes n denson x CM x x February 7, 08

47 Center o Mass or any partcles n 3D? February 7, 08

48 Where s the Center o Mass? Assue = kg, = 3 kg, and x =, x = 5, where s the center o ass o these two objects? x A) x CM = B) x CM = C) x CM = 3 D) x CM = 4 E) x CM = 5 x CM February 7, 08 x

49 Center o Mass or a Syste o Partcles Two bodes and one denson General case: n bodes and three denson where M = February 7, 08

50 Saple Proble : Three partcles o asses =. kg, =.5 kg, and 3 = 3.4 kg or an equlateral trangle o edge length a = 40 c. Where s the center o ass o ths syste? (Hnt: s at (0,0), s at (40 c,0), and 3 s at (70 c, 0 c), as shown n the gure below.) x CM M n x x x 3x 3 3 y CM M n y y y 3 3 y 3 x CM 8.8 c and y CM 57.5 c February 7, 08

51 Moton o a Syste o Partcles Assue the total ass, M, o the syste reans constant We can descrbe the oton o the syste n ters o the elocty and acceleraton o the center o ass o the syste We can also descrbe the oentu o the syste and Newton s Second Law or the syste February 7, 08

52 Velocty and Moentu o a Syste o Partcles The elocty o the center o ass o a syste o partcles s CM dr CM dt M The oentu can be expressed as CM The total lnear oentu o the syste equals the total ass ultpled by the elocty o the center o ass M p p tot February 7, 08

53 Acceleraton and Force o the Center o Mass The acceleraton o the center o ass can be ound by derentatng the elocty wth respect to te a CM d CM dt M The acceleraton can be related to a orce Ma CM I we su oer all the nternal orces, they cancel n pars and the net orce on the syste s caused only by the external orces F a February 7, 08

54 Newton s Second Law or a Syste o Partcles Snce the only orces are external, the net external orce equals the total ass o the syste ultpled by the acceleraton o the center o ass: F ext Ma CM The center o ass o a syste o partcles o cobned ass M oes lke an equalent partcle o ass M would oe under the nluence o the net external orce on the syste February 7, 08