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1 Physcs 0 Lecture 9 Lnear Momentum and Collsons Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department
2 Lnear Momentum q Conseraton o Energy q Momentum q Impulse q Conseraton o Momentum q D Collsons q D Collsons q The Center o Mass and Collsons
3 Conseraton o Energy q D E = D K + D U = 0 conserate orces are the only orces that do work on the system. q The total amount o energy n the system s constant. m + mgy + kx = m + mgy + kx q D E = D K + D U =  k d rcton orces are dong work on the system. q The total amount o energy n the system s stll constant, but the change n mechancal energy goes nto nternal energy or heat.  k d = æ ç è m + mgy + kx ö  ø æ ç è m + mgy + kx ö ø
4 Lnear Momentum q Ths s a new undamental quantty, lke orce, energy. It s a ector quantty (ponts n same drecton as elocty). q The lnear momentum p o an object o mass m mong wth a elocty s dened to be the product o the mass and elocty: p = m q The terms momentum and lnear momentum wll be used nterchangeably n the text q Momentum depend on an object s mass and elocty
5 Lnear Momentum q Lnear momentum s a ector quantty p= m n Its drecton s the same as the drecton o the elocty q The dmensons o momentum are ML/T q The SI unts o momentum are kg m / s q Momentum can be expressed n component orm: p x = m x p y = m y p z = m z
6 Newton s Law and Momentum q Newton s Second Law can be used to relate the momentum o an object to the resultant orce actng on t D D( m) F net = ma = m = Dt Dt q The change n an object s momentum dded by the elapsed tme equals the constant net orce actng on the object p D change n momentum = = Dt tme nteral F net
7 Impulse q When a sngle, constant orce acts on the object, there s an mpulse delered to the object n n I = FDt s dened as the mpulse I n The equalty s true een the orce s not constant n Vector quantty, the drecton s the same as the drecton o the orce p D change n momentum = = F net Dt tme nteral
8 ImpulseMomentum q The theorem states that the mpulse actng on a system s equal to the change n momentum o the system Dp = FnetDt = I I = Dp = m  m Theorem
9 Calculatng the Change o D p= p  p ater Momentum beore = m  ater m beore = m (  ) ater For the teddy bear [ 0 ( )] beore D p= m  = m For the bouncng ball [ ] D p=  ( ) = m
10 Ex: How Good Are the Bumpers? q In a crash test, a car o mass kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are =5 m/s and =.6 m/s, respectely. I the collson lasts or 0.5 s, nd (a) the mpulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car
11 How Good Are the Bumpers? q In a crash test, a car o mass kg colldes wth a wall and rebounds as n gure. The ntal and nal eloctes o the car are =5 m/s and =.6 m/s, respectely. I the collson lasts or 0.5 s, nd (a) the mpulse delered to the car due to the collson (b) the sze and drecton o the aerage orce exerted on the car p p = m = m = ( = (.5 0 kg)( 5m / s) = kg)( +.6m / s) = kg m / s 4 kg m/ s I = Dp Dt p  I = Dt p = ( = = m 4 kg m / s)  (.5 0 kg m / s  m kg m / s 0.5s.76 kg m / s) 0 5 F a = = = 4 N
12 Ex: ImpulseMomentum q Theorem A chld bounces a 00 g superball on the sdewalk. The elocty o the superball changes rom 0 m/s downward to 0 m/s upward. I the contact tme wth the sdewalk s 0.s, what s the magntude o the mpulse mparted to the superball? (A) 0 (B) kgm/s (C) 0 kgm/s (D) 00 kgm/s (E) 000 kgm/s I = Dp = m  m
13 Ex3: ImpulseMomentum q Theorem A chld bounces a 00 g superball on the sdewalk. The elocty o the superball changes rom 0 m/s downward to 0 m/s upward. I the contact tme wth the sdewalk s 0.s, what s the magntude o the orce between the sdewalk and the superball? (A) 0 (B) N I Dp m  F = = = (C) 0 N Dt Dt Dt (D) 00 N (E) 000 N m
14 Conseraton o Momentum q In an solated and closed system, the total momentum o the system remans constant n tme. n Isolated system: no external orces n Closed system: no mass enters or leaes n The lnear momentum o each colldng body may change n The total momentum P o the system cannot change.
15 Conseraton o Momentum q Start rom mpulsemomentum theorem Dt =  m F FDt = m  m q Snce q Then q So F t = F D Dt  m = ( m  m + = m m m m + )
16 Conseraton o Momentum q When no external orces act on a system consstng o two objects that collde wth each other, the total momentum o the system remans constant n tme F Dt = Dp = p  p net q When F net = 0 then q For an solated system p = p Dp = 0 q Speccally, the total momentum beore the collson wll equal the total momentum ater the collson + = m m m m +
17 Ex4: The Archer q An archer stands at rest on rctonless ce and res a 0.5kg arrow horzontally at 50.0 m/s. The combned mass o the archer and bow s 60.0 kg. Wth what elocty does the archer moe across the ce ater rng the arrow? p = p + m = m + m m = 60.0kg, m = 0.5kg, = = 0, = 50m / s, = 0 = + m m? m 0.5kg =  =  (50.0m / s) = 0.47m / m 60.0kg s
18 Ex5: Conseraton o Momentum q A 00 kg man and 50 kg woman on ce skates stand acng each other. I the woman pushes the man backwards so that hs nal speed s m/s, at what speed does she recol? (A) 0 (B) 0.5 m/s (C) m/s (D).44 m/s (E) m/s
19 Types o Collsons q Momentum s consered n any collson q Inelastc collsons: rubber ball and hard ball n Knetc energy s not consered n Perectly nelastc collsons occur when the objects stck together q Elastc collsons: bllard ball n both momentum and knetc energy are consered
20 Collsons Summary q In an elastc collson, both momentum and knetc energy are consered q In a nonperect nelastc collson, momentum s consered but knetc energy s not. Moreoer, the objects do not stck together q In a perectly nelastc collson, momentum s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the same q Elastc and perectly nelastc collsons are lmtng cases, most actual collsons all n between these two types q Momentum s consered n all collsons
21 More about Perectly Inelastc Collsons q When two objects stck together ater the collson, they hae undergone a perectly nelastc collson q Conseraton o momentum m ) + m = ( m + m = m m + + m m q Knetc energy s NOT consered
22 Ex6: An SUV Versus a Compact q An SUV wth mass kg s traellng eastbound at +5.0 m/s, whle a compact car wth mass kg s traellng westbound at 5.0 m/s. The cars collde headon, becomng 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 system consstng o both cars.
23 An SUV Versus a Compact (a) Fnd the speed o the entangled cars ater the collson. p = p m ) + m = ( m + m m m = = kg, kg, = + 5m / s = 5m / s = m m + + m m = +5.00m / s
24 An SUV Versus a Compact (b) Fnd the change n the elocty o each car. D D = +5.00m / =  s  = 0.0m / = +  = 0.0m / s s m m = = kg, kg, = + 5m / s = 5m / s 4 m D = m (  ) = kg m/ s 4 m D = m (  ) = kg m s / m D + md = 0
25 An SUV Versus a Compact (c) Fnd the change n the knetc energy o the system consstng o both cars. = +5.00m / s m m = = kg, kg, = + 5m / s = 5m / s KE KE 5 = m + m = J 4 = m + m = J DKE = KE  KE = J
26 More About Elastc Collsons q Both momentum and knetc energy are consered q Typcally hae two unknowns q Momentum s a ector quantty n n + + = = Drecton s mportant Be sure to hae the correct sgns q Sole the equatons smultaneously + +
27 Elastc Collsons q A smpler equaton can be used n place o the KE equaton + = + ) (  =   m m + = + ) )( ( ) )( ( m m +  = +  ) ( ) ( m m  =  ) ( ) ( m m  =  m m + = + m m + = +
28 Summary o Types o Collsons q In an elastc collson, both momentum and knetc energy are consered + = + + m = m + m q In an nelastc collson, momentum s consered but knetc energy s not + = m + q In a perectly nelastc collson, momentum s consered, knetc energy s not, and the two objects stck together ater the collson, so ther nal eloctes are the same m ) + m = ( m + m
29 Ex7: Conseraton o q Momentum An object o mass m moes to the rght wth a speed. It colldes headon wth an object o mass 3m mong wth speed /3 n the opposte drecton. I the two objects stck together, what s the speed o the combned object, o mass 4m, ater the collson? (A) 0 (B) / (C) (D) (E) 4
30 Problem Solng or D Collsons, q Coordnates: Set up a coordnate axs and dene the eloctes wth respect to ths axs n It s conenent to make your axs concde wth one o the ntal eloctes q Dagram: In your sketch, draw all the elocty ectors and label the eloctes and the masses
31 Problem Solng or D Collsons, q Conseraton o Momentum: Wrte a general expresson or the total momentum o the system beore and ater the collson n Equate the two total momentum expressons n Fll n the known alues + m = m + m
32 Problem Solng or D Collsons, 3 q Conseraton o Energy: I the collson s elastc, wrte a second equaton or conseraton o KE, or the alternate equaton n Ths only apples to perectly elastc collsons + = + q Sole: the resultng equatons smultaneously
33 OneDmenson s Two Dmenson
34 TwoDmensonal Collsons q For a general collson o two objects n twodmensonal space, the conseraton o momentum prncple mples that the total momentum o the system n each drecton s consered x + x = x + x y + y = y + y
35 TwoDmensonal Collsons q The momentum s consered n all drectons q Use subscrpts or n Identyng the object m n Indcatng ntal or nal alues n The elocty components q I the collson s elastc, use conseraton o knetc energy as a second equaton n Remember, the smpler equaton can only be used or onedmensonal stuatons x y + + x y = = x y = + x y
36 Glancng Collsons q The ater eloctes hae x and y components q Momentum s consered n the x drecton and n the y drecton q Apply conseraton o momentum separately to each drecton x y + + x y = = x y + + x y
37 D Collson, example q Partcle s mong at elocty and partcle s at rest q In the xdrecton, the ntal momentum s m q In the ydrecton, the ntal momentum s 0
38 D Collson, example cont q Ater the collson, the momentum n the xdrecton s m cos q + m cos q Ater the collson, the momentum n the ydrecton s m sn q + m sn = + 0 = cosq + sn q  sn cos q I the collson s elastc, apply the knetc energy equaton m = m + m
39 Ex8: Collson at an Intersecton q A car wth mass kg traelng east at a speed o 5 m/s colldes at an ntersecton wth a kg an traelng north at a speed o 0 m/s. Fnd the magntude and drecton o the elocty o the wreckage ater the collson, assumng that the ehcles undergo a perectly nelastc collson and assumng that rcton between the ehcles and the road can be neglected. m c cx = = 5m / s, kg, m y =.5 0 = 0m / s, 3 kg =? q =?
40 m c cx Collson at an Intersecton = = 5 m/s, kg, m y =.5 0 = 0 m/s, 3 kg =? q =? 4 å px = mccx + mx = mccx = kg m/s å px = mccx + mx = ( mc + m ) cosq kg m/s = ( kg) cosq 4 å py = mccy + my = my = kg m/s å py = mccy + my = ( mc + m ) snq kg m/s = ( kg) snq
41 m c cx Collson at an Intersecton = = 5m / s, 4 4 kg, m y =.5 0 = 0m / s, kg m/s = ( kg m/s = ( kg) kg) 3 kg =? q =? snq cosq kg m/ s tanq = kg m/ s =.33  q = tan (.33) = kg m/s 3 ( kg)sn53. = = 5.6 m/s
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