Physcs 0 Lecture 9 Lnear Momentum and Collsons Dr. Al ÖVGÜN EMU Physcs Department www.aogun.com
Lnear Momentum and Collsons q q q q q q q Conseraton o Energy Momentum Impulse Conseraton o Momentum -D Collsons -D Collsons The Center o Mass February 3, 07
Conseraton o Energy q Δ E = Δ K + Δ 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 Δ E = Δ K + Δ 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 = m + mgy + kx m + mgy + kx February 3, 07
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 February 3, 07
Lnear Momentum q Lnear momentum s a ector quantty 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 r p r = m February 3, 07
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 Δ Δ( m) F net = ma = m = Δt Δt q The change n an object s momentum dded by the elapsed tme equals the constant net orce actng on the object p Δ change n momentum = = Δt tme nteral F net February 3, 07
Impulse q When a sngle, constant orce acts on the object, there s an mpulse delered to the object n n I r = F Δt 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 Δ change n momentum = = F net Δt tme nteral February 3, 07
Impulse-Momentum q The theorem states that the mpulse actng on a system s equal to the change n momentum o the system Δp = FnetΔt = I I = Δp = m m Theorem February 3, 07
Calculatng the Change o r r r Δ p= p p ater Momentum beore = m ater m beore = m ( ) ater For the teddy bear [ 0 ( )] beore Δ p= m = m For the bouncng ball [ ] Δ p= m ( ) = m February 3, 07
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 February 3, 07
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 = Δp Δt p I Δt 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 kg m / s) 5 F a = = = =.76 0 4 N February 3, 07
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 = Δp = m m February 3, 07
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 Δp m F = = = (C) 0 N Δt Δt Δt (D) 00 N (E) 000 N m February 3, 07
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. February 3, 07
Conseraton o Momentum q Start rom mpulse-momentum theorem Δt = m m F FΔt = m m q Snce q Then q So m F t = F Δ Δt m = ( m m m + m = m + m ) February 3, 07
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 Δt = Δp = p p net q When F net = 0 then q For an solated system p = p Δp = 0 q Speccally, the total momentum beore the collson wll equal the total momentum ater the collson m + m = m + m February 3, 07
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 m = 60.0kg, m = 0.5kg, = = 0, = 50m / s, = 0 = + m m? m 0.5kg = = (50.0m / s) = 0.47m / m 60.0kg s February 3, 07
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 February 3, 07
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 February 3, 07
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 February 3, 07
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 February 3, 07
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. February 3, 07
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 February 3, 07
An SUV Versus a Compact (b) Fnd the change n the elocty o each car. Δ Δ = +5.00m / = s = 0.0m / = + = 0.0m / s s m m =.80 0 3 = 9.00 0 kg, kg, = + 5m / s = 5m / s 4 mδ = m ( ) =.8 0 kg m/ s 4 m Δ = m ( ) = +.8 0 kg m s / m Δ + mδ = 0 February 3, 07
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 ΔKE = KE KE =.70 0 5 J February 3, 07
More About Elastc Collsons q Both momentum and knetc energy are consered m q Typcally hae two unknowns q Momentum s a ector quantty n n m + m + = m m = Drecton s mportant Be sure to hae the correct sgns q Sole the equatons smultaneously + m m + m February 3, 07
February 3, 07 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 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 + m q In an nelastc collson, momentum s consered but knetc energy s not + m = m m 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 February 3, 07
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 February 3, 07
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 February 3, 07
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 + m February 3, 07
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 February 3, 07
One-Dmenson s Two- Dmenson February 3, 07
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 m x + m x = m x + m x m y + m y = m y + m y February 3, 07
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 m 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 + m + m x y = = m m x y + m + m + = + x y February 3, 07
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 m m x y + m + m x y = m = m x y + m + m x y February 3, 07
-D Collson, example q Partcle s mong at elocty r 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 February 3, 07
-D Collson, example cont q Ater the collson, the momentum n the x-drecton s m cos θ + m cos φ q Ater the collson, the momentum n the y-drecton s m sn θ + m sn φ m 0 + 0 = + 0 = m m cosθ + m sn θ m sn φ cosφ q I the collson s elastc, apply the knetc energy equaton m = m + m February 3, 07
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 =? θ =? February 3, 07
p x m c cx Collson at an Intersecton =.5 0 3 = 5 m/s, kg, m y =.5 0 = 0 m/s, 3 kg =? θ =? 4 = m + m = m = 3.75 0 kg m/s c cx x px = mccx + mx = ( mc + m ) cosθ 3.75 0 p y 4 c cx kg m/s = (4.00 0 3 kg) cosθ 4 = m + m = m = 5.00 0 kg m/s c cy y py = mccy + my = ( mc + m ) snθ 5.00 0 4 y kg m/s = (4.00 0 3 kg) snθ February 3, 07
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 =? θ =? snθ cosθ 4 5.00 0 kg m/ s tanθ = 4 3.75 0 kg m/ s =.33 θ = tan (.33) = 53. 4 5.00 0 kg m/s 3 (4.00 0 kg)sn53. = = 5.6 m/s February 3, 07
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