Physcs 0 Lecture 9 Lnear Momentum and Collsons Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aogun.com
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
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
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
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
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
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
Impulse-Momentum 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
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
Ex: How Good Are the Bumpers? q In a crash test, a car o mass.5 0 3 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
How Good Are the Bumpers? q In a crash test, a car o mass.5 0 3 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 3 = (.5 0 kg)( -5m / s) = -.5 0 3 kg)( +.6m / s) = + 0.39 0 4 kg m / s 4 kg m/ s I = Dp Dt p - I = Dt p = (0.39 0 =.64 0 4 = m 4 kg m / s) - (-.5 0 kg m / s - m 4.64 0 kg m / s 0.5s.76 kg m / s) 0 5 F a = = = 4 N
Ex: Impulse-Momentum 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) kg-m/s (C) 0 kg-m/s (D) 00 kg-m/s (E) 000 kg-m/s I = Dp = m - m
Ex3: Impulse-Momentum 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
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.
Conseraton o Momentum q Start rom mpulse-momentum 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 + )
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 +
Ex4: The Archer q An archer stands at rest on rctonless ce and res a 0.5-kg 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
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
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
Collsons Summary q In an elastc collson, both momentum and knetc energy are consered q In a non-perect 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
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
Ex6: An SUV Versus a Compact q An SUV wth mass.80 0 3 kg s traellng eastbound at +5.0 m/s, whle a compact car wth mass 9.00 0 kg s traellng westbound at -5.0 m/s. The cars collde head-on, 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.
An SUV Versus a Compact (a) Fnd the speed o the entangled cars ater the collson. p = p m ) + m = ( m + m m m =.80 0 3 = 9.00 0 kg, kg, = + 5m / s = -5m / s = m m + + m m = +5.00m / s
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 =.80 0 3 = 9.00 0 kg, kg, = + 5m / s = -5m / s 4 m D = m ( - ) = -.8 0 kg m/ s 4 m D = m ( - ) = +.8 0 kg m s / m D + md = 0
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 =.80 0 3 = 9.00 0 kg, kg, = + 5m / s = -5m / s KE KE 5 = m + m = 3.04 0 J 4 = m + m = 3.38 0 J DKE = KE - KE = -.70 0 5 J
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 + +
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 + = +
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
Ex7: Conseraton o q Momentum An object o mass m moes to the rght wth a speed. It colldes head-on 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
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
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
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
One-Dmenson s Two- Dmenson
Two-Dmensonal 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
Two-Dmensonal 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 one-dmensonal stuatons x y + + x y = = x y + + + = + x y
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
-D Collson, example q Partcle s mong at elocty and partcle s at rest q In the x-drecton, the ntal momentum s m q In the y-drecton, the ntal momentum s 0
-D Collson, example cont q Ater the collson, the momentum n the x-drecton s m cos q + m cos q Ater the collson, the momentum n the y-drecton s m sn q + m sn 0 + 0 = + 0 = cosq + sn q - sn cos q I the collson s elastc, apply the knetc energy equaton m = m + m
Ex8: Collson at an Intersecton q A car wth mass.5 0 3 kg traelng east at a speed o 5 m/s colldes at an ntersecton wth a.5 0 3 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 =.5 0 3 = 5m / s, kg, m y =.5 0 = 0m / s, 3 kg =? q =?
m c cx Collson at an Intersecton =.5 0 3 = 5 m/s, kg, m y =.5 0 = 0 m/s, 3 kg =? q =? 4 å px = mccx + mx = mccx = 3.75 0 kg m/s å px = mccx + mx = ( mc + m ) cosq 3.75 0 4 kg m/s = (4.00 0 3 kg) cosq 4 å py = mccy + my = my = 5.00 0 kg m/s å py = mccy + my = ( mc + m ) snq 5.00 0 4 kg m/s = (4.00 0 3 kg) snq
m c cx 5.00 0 3.75 0 Collson at an Intersecton =.5 0 3 = 5m / s, 4 4 kg, m y =.5 0 = 0m / s, kg m/s = (4.00 0 kg m/s = (4.00 0 3 3 kg) kg) 3 kg =? q =? snq cosq 4 5.00 0 kg m/ s tanq = 4 3.75 0 kg m/ s =.33 - q = tan (.33) = 53. 4 5.00 0 kg m/s 3 (4.00 0 kg)sn53. = = 5.6 m/s
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